Working Paper 25420http://www.nber.org/papers/w25420
NATIONAL BUREAU OF ECONOMIC RESEARCH1050 Massachusetts Avenue
Cambridge, MA 02138January 2019
We acknowledge many fruitful suggestions by Raymond Kan and discussions with Gurdip Bakshi, Utpal Bhattacharya, Steve Riddiough, Giorgio Valente and Jialin Yu, who also helped us obtain some of the data used in the paper. Participants at the 2016 Annual Conference in International Finance at City University of Hong Kong, the Bank of England, Banca d’Italia and ECB 6-thWorkshop on Financial Determinants of Foreign Exchange Rates, and seminars at Nanyang Technology University, UNSW, SAIF and Xiamen University provided helpful insights. Any remaining errors are our responsibility alone. Panayotov acknowledges support from a Hong Kong RGC Research grant (No. 16505715). The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research.
NBER working papers are circulated for discussion and comment purposes. They have not been peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications.
Good Carry, Bad CarryGeert Bekaert and George PanayotovNBER Working Paper No. 25420January 2019JEL No. C23,C53,G11
ABSTRACT
We distinguish between ”good” and ”bad” carry trades constructed from G-10 currencies. The good trades exhibit higher Sharpe ratios and sometimes positive return skewness, in contrast to the bad trades that have both substantially lower Sharpe ratios and highly negative return skewness. Surprisingly, good trades do not involve the most typical carry currencies like the Australian dollar and Japanese yen. The distinction between good and bad carry trades significantly alters our understanding of currency carry trade returns, and invalidates, for example, explanations invoking return skewness and crash risk.
Geert BekaertGraduate School of BusinessColumbia University3022 Broadway, 411 Uris HallNew York, NY 10027and [email protected]
George PanayotovHong Kong University of Science and TechnologyBusiness School - Dept. of FinanceClear Water BayKowloon, [email protected]
I Introduction
The currency carry trade, which goes long (short) currencies with high (low) yields continues to attract
much research attention, as it has been shown to earn high Sharpe ratios, while its returns are largely un-
correlated with standard systematic risks (e.g., Burnside, Eichenbaum, Kleshchelski, and Rebelo (2011),
Lustig, Roussanov, and Verdelhan (2011)). Prototypical carry currencies among the liquid G-10 curren-
cies are the Swiss franc (CHF) and Japanese yen (JPY), which almost always exhibit the lowest yields
and hence a typical G-10 carry trade would short them, and the New Zealand dollar (NZD) and Australian
dollar (AUD), which typically have the highest yields and would be held long. These four currencies
feature prominently in extant explanations of the returns of the carry trade. One such explanation invokes
crash risk (e.g. Brunnermeier, Nagel, and Pedersen (2009)) and thus relies implicitly on the fact that JPY
and AUD provide the most ”skewed” return perspectives from the view point of a US investor. Another is
based on the differential exposure to global productivity shocks of producers of final goods, such as Japan
and Switzerland, versus commodity producers, such as Australia and New Zealand (Ready, Roussanov,
and Ward (2017)).
While the prior literature takes for granted that the prototypical carry currencies1 drive carry trade
profitability, we document the existence of ”good” and ”bad” currency carry trades. We consider an
investor who sequentially tests whether reducing the set of G-10 currencies improves the historical Sharpe
ratio, and then implements equally weighted carry trades with fewer currencies. We find that such trades
improve the return profile (in terms of both Sharpe ratio and skewness) relative to the carry trade which
employs all G-10 currencies, and denote them as ”good” carry trades. Most surprisingly, these good trades
almost never include the AUD and JPY, or the NOK - another commodity currency. Next, we construct
1To clarify terminology, we note that a key component of the return of a carry trade is the interest rate (or forward) dif-ferential between the investment and funding currencies that are long and short in the trade, respectively. This component isoften, even if perhaps confusingly, referred to as ”carry”, and we also follow this tradition, for brevity. For the same reason wealso sometimes refer to ”carry currencies”, ”carry profitability”, ”carry returns”, etc., instead of ”carry trade currencies”, ”carrytrade profitability” and ”carry trade returns”. The context is clear in all these cases, and should allow no ambiguity.
1
carry trades using fixed subsets of the G-10 currencies over the full sample, and find that trades involving
only the prototypical currencies have lower Sharpe ratios and more negatively skewed returns. We denote
them as ”bad” carry trades. The trades using the remaining currencies preserve the desirable features of
”good” carry trades.
Providing a first glimpse on the issue, Figure 1 contrasts the return properties of carry trades that in-
volve various subsets of the G-10 currencies. In particular, the figure plots (with black dots) the skewness
versus Sharpe ratio for all carry trades constructed from five currencies that use three of the prototypical
currencies (AUD, CHF and JPY), together with any possible pair from the remaining seven currencies.
The currencies enter each trade with equal weights, as is common in the literature and finance industry.
Strikingly, these 21 trades show worse Sharpe ratios, and also substantially lower skewness than the strat-
egy that uses all G-10 currencies (denoted with the horizontal and vertical lines in the graph). Therefore,
trades constructed predominantly from the prototypical carry currencies appear to be ”bad” carry trades.
We subsequently refer to the trade from all G-10 currencies as ”standard carry” and denote it as SC.
Probing further, Figure 1 also displays (with unfilled circles) the skewness versus Sharpe ratio of the
complements of the previous 21 carry trades, which are constructed with the remaining five currencies in
each case, again with equal weights. It is noteworthy that 14 out of the 21 complement trades feature higher
Sharpe ratios than that of the standard carry (SC) trade (in one case almost double that ratio), and 16 show
higher (less negative or positive) skewness. Furthermore, half of the complement trades improve both on
the skewness and Sharpe ratio of the SC trade, qualifying them as ”good” carry trades. These findings
cast doubt on efforts to explain carry trade returns by focusing on properties of the prototypical carry
currencies, and undermine the practice of associating the carry trade predominantly with such currencies.
In Section IV we investigate the ability of good carry trades to function as risk factors for certain cross
sections of currency returns (see, e.g. Lustig, Roussanov, and Verdelhan (2011) and Menkhoff, Sarno,
2
Schmeling, and Schrimpf (2012)). We find that good carry trades perform at least as well as previously
suggested currency market risk factors, and sometimes drive out such factors in a horse race. We also
re-examine the predictability findings in Bakshi and Panayotov (2013) and Ready et al. (2017), and find
that previously identified carry return predictors strongly predict the returns of bad, but not of good carry
trades. In Section V we revisit several interpretations of carry returns that have been advanced in the recent
literature, including the explanatory ability of factor models with equity market risk factors, a crash risk
explanation of their returns as in Brunnermeier et al. (2009), and the peso problem hypothesis of Burnside
et al. (2011). Almost invariably, the results differ greatly across good versus bad carry trades.
In Section VI, we further explore the properties of good and bad carry trades to kindle research on
economic models that may explain the strong differences between the two types of trades. We show, for
example, that the returns of bad (good) trades derive mostly from the gain that comes from investing at
the higher interest rate while borrowing at the lower interest rate, which is partially offset (reinforced)
by exchange rate changes. We also examine the relationship between good carry trades and the ”dollar
carry” trade introduced in Lustig, Roussanov, and Verdelhan (2014), which goes long (short) all currencies
relative to the US dollar when the average foreign interest rate differential relative to the dollar is positive
(negative). Because our good carry trades always involve the dollar, they do show substantial return
correlation with dollar carry, but we demonstrate that they clearly present a distinct currency and economic
risk.
Before introducing ”good” and ”bad” carry trades in Section 3, we describe the data in Section 2
and discuss some important concepts regarding the design of carry trades. The remainder of the article
demonstrates how the good-bad trade distinction fundamentally alters our thinking about carry trades.
3
II Data and carry trade design
Following previous work, we employ currency spot and forward contract quotes to construct carry
trade returns. Using one-month forward quotes on the last trading day of each month in the sample, and
spot quotes on the last day of the following month, we calculate one-month carry trade returns over the
sample period from 12/1984 till 06/2014 (354 monthly observations). The return calculations take into
account transaction costs, exploiting the availability of bid and ask quotes. The data comes from Barclays
Bank, as available on Datastream, and have been used in Burnside et al. (2011) and Lustig at al. (2011,
2014), among many others.
Our results are reported for percentage returns and equal (absolute) weights of the currencies entering
a trade. On two occasions we report instead results with logarithmic returns or weights proportional to
forward differentials, to facilitate comparability with previous studies.
Let Sit (F i
t ) denote the spot (forward) exchange rate of currency i at time t, quoted as foreign currency
units per U.S. dollar. That is, the U.S. dollar is the benchmark currency and all trades are implemented
relative to the dollar. Then, with t indicating the end of a given month, the percentage excess one-month
return at t +1 of one dollar invested at t in a long (short) forward foreign currency contract is:
Rxi,longt+1 = F i,bid
t /Si,askt+1 −1 and Rxi,short
t+1 = 1−F i,askt /Si,bid
t+1 ,(1)
whereby bid and ask quotes are denoted in the superscript.2
We employ the G-10 currencies, which are the New Zealand dollar (NZD), Australian dollar (AUD),
British pound (GBP), Norwegian krone (NOK), Swedish krona (SEK), Canadian dollar (CAD), US dollar
(USD), Euro (EUR), Swiss franc (CHF) and Japanese yen (JPY), whereby prior to 1999 the German mark
(DEM) is used instead of the Euro. These currencies represent the most liquid traded currencies, and are
most often used both in the academic literature and professional practice to construct carry trades.
2These can also be seen as the payoffs to forward contracts in the foreign currency per ”forward” dollar.
4
As indicated above, the carry trades that we consider go long and short an equal number of currencies
relative to the USD, with equal weights. Various alternative weighting schemes are possible, mostly based
on the magnitude of the interest rate differentials (see Table 1 for concrete examples from practice and the
academic literature), but we prefer to keep the trade as simple as possible. Moreover, the total investment
each period is one dollar, that is, the sum of all long and short positions (in absolute value) equals one.
Specifically, when the trade uses all G-10 currencies, the five currencies with the lowest interest rates at
the end of each month are shorted, and the remaining five are held long. In practice, we rank the currencies
based on their forward differentials relative to the U.S. dollar, defined as FDt = Ft/St − 1 at time t and
calculated using mid-quotes. The weight of currency i held long (short) is ωit =
110 (ωi
t = − 110 ). The
percentage excess return of this trade from t to t +1 is:
Rxcarryt+1 =
10
∑i=1
{1
ωit>0ω
it Rxi,long
t+1 −1ωi
t<0ωit Rxi,short
t+1
}(2)
where 1(.) is an indicator function.
When a subset of N currencies is used to construct a carry trade and N is even, we set ωit =
1N or − 1
N
in (2) and substitute N for 10. If N is odd, the currency with the median forward differential is dropped
from the trade, and we use N−1 instead of N in the definition of ωit and the summation in (2).
We consider carry trades that are symmetric, in that they have an equal number of short and long
positions, with equal total weights on the long and short side. Currencies are ranked according to their
interest rates, and only the rank determines whether the position taken is short or long, while the signs
of the interest rate differentials are irrelevant for the trade design, as these change with the currency per-
spective (see also Clarida, Davis, and Pedersen (2009)). Importantly, our carry trade design also ensures
(approximate) numeraire independence, as we do not give a special role to the benchmark currency, and
hence the positions taken in the various participating currencies are the same, regardless of the bench-
5
mark. Numeraire independence is an attractive property, and implies that only one currency trade must
be defined for the world at large. Moreover, the returns on such a trade are very similar from any cur-
rency perspective, because the translation from one currency to another simply introduces cross-currency
risk on currency returns, which is a second-order effect. In fact, the logarithmic returns of our strategies
are exactly the same from any perspective, by triangular arbitrage (see Maurer, To, and Tran (2018) for
further discussion). The major commercial investable carry products delivered by the major players in
the foreign exchange market, such as Deutsche Bank or Citibank, described in Table 1, are symmetric
and numeraire-independent as per our definition. They do not all assign equal weights to all positions
however, e.g. the well-known tradeable Deutsche Bank carry strategy takes only the three highest- and
lowest-yielding currencies among the G-10 currencies.
Non-symmetric trade designs are also possible, and have been considered, for example in a recent well-
recognized article by Burnside et al. (2011), where all currencies with interest rates that are higher (lower)
than the US dollar interest rate are bought (sold) in equal proportions. Such a strategy is obviously not
symmetric, and also may deliver very different results depending on the benchmark currency (see Daniel,
Hodrick, and Lu (2017) for further discussion). Another example of a non-symmetric trade is the ”dollar
carry” trade, studied in Lustig et al. (2014). These trades are also ”dollar-neutral”, excluding positions
in the benchmark currency (which would generate zero excess returns). In contrast, our symmetric trades
are not dollar-neutral: positions in the benchmark currency are explicitly included.
III Good and bad carry trades from the G-10 currencies
In this section we first outline a disciplined approach to create historically attractive symmetric carry
trades from subsets of the G-10 currencies, exploiting all available foreign exchange history at each point
in time. It evaluates on each trading date whether excluding currencies can improve on the standard carry
6
trade (SC) that uses all G-10 currencies, and if so, which currencies should be excluded. Because of its
dynamic implementation, the procedure yields out-of-sample results. Next, we exploit the information
garnered in his exercise to create fixed subsets of ”good” and ”bad” currencies that do not change over
time, and use them to construct carry trades over the full sample period.
A Enhancing the currency carry trade
Imagine an investor starting to trade at the end of December 1994 (t = T1). On this date, and at the end
of each month going forward till May 2014 (t = T2), he uses all available return information for the period
since December 1984 (t = T0) and first calculates the Sharpe ratio (denoted ”benchmark Sharpe ratio”) of
the standard carry trade (SC) that employs all G-10 currencies. The trade ranks these currencies according
to their forward differentials at the end of each month between T0 and t− 1, for t = T1, ...,T2, and goes
long (short) over the following month the five currencies with the highest (lowest) forward differentials,
all with equal weights.
To create an enhanced carry trade with nine currencies, on date t the investor excludes one by one
each of the G-10 currencies, and computes the Sharpe ratios over T0 to t of the ten possible trades that
involve only nine currencies. These trades exclude the currency with the median forward differential at the
end of each month between T0 and t−1 and go long (short) the four currencies with the highest (lowest)
differentials. If the highest of the ten Sharpe ratios obtained in this way exceeds the benchmark Sharpe
ratio, an enhanced trade is implemented over the following month (t to t + 1) using the nine currencies
corresponding to this highest Sharpe ratio, while the one currency left out of the trade is the first to be
excluded on date t. If, on the other hand, all ten Sharpe ratios are lower than the benchmark ratio, then no
currency is excluded and the enhanced trade for this date has the return of the standard trade.
Note that the dynamic and real-time nature of this enhanced trade could, in principle, result in a
substantially different currency mix used at different points of time. Further, our enhancement rule is
7
intentionally simple and uses an easily understood and popular performance measure, whereas a wide
range of sophisticated optimization rules could be applied in this context as well (see, e.g., Barroso and
Santa-Clara (2014)). Mimicking the construction of the enhanced trade that uses nine currencies, we
construct analogous enhanced trades that exclude more than one of the G-10 currencies. In particular,
on date t when one currency has been excluded, we use the remaining nine currencies to find the highest
Sharpe ratio across the nine possible trades that involve only eight currencies. Again, if this highest ratio
exceeds the benchmark Sharpe ratio, the currency that was omitted to achieve it is the second currency
to be excluded for this date, whereas if all Sharpe ratios are lower than the benchmark one, no further
currency is excluded and the enhanced trade that uses eight currencies has the return of the standard trade
for that date. Similarly, we attempt to exclude up to seven of the G-10 currencies on this date t, and thus
obtain seven enhanced trades, which use a decreasing number of currencies. Importantly, we record the
exact order in which currencies have been excluded. The above procedure is repeated on each date in the
sample to obtain time series of returns of the seven enhanced carry trades.
For completeness of the search algorithm, we have postulated that if no improvement on the bench-
mark Sharpe ratio can be achieved for a certain date and number of excluded currencies, then no further
currencies are excluded on this date, and all enhanced trades with fewer currencies have the return of the
standard trade. In practice, however, this choice is inconsequential (see below).
B Return patterns for enhanced trades
Table 2 presents results for the enhanced carry trades that allow excluding from one to as much as
seven currencies on each trading date, and, for comparability, for the standard carry trade. Returns are
computed as described in Section II, with equal currency weights and in percent. Panel A of Table 2
reports the annualized average returns, annualized Sharpe ratios, and return skewness for each carry trade,
with interest in skewness justified given its important role in certain explanations for the carry trade returns
8
(e.g., Brunnermeier et al. (2009), Jurek (2014)). Also reported are p-values for the hypothesis that the
respective Sharpe ratio or skewness does not exceed that for the standard trade SC. Because carry returns
tend to be negatively skewed, we cannot rely on standard tests for the difference between Sharpe ratios,
as for example in Jobson and Korkie (1981) or Memmel (2003), that apply to Gaussian distributions.
Therefore, we resort to the bootstrap tests, described in Ledoit and Wolf (2008) for Sharpe ratios, and
Annaert, Van Osselaer, and Verstraete (2009) for skewness (see Appendix OA-II for details). Our p-
values use one-sided bootstrap confidence intervals, as the enhanced carry trades are designed with the
goal to improve on SC.
The SC trade has an annualized Sharpe ratio of 0.32, and return skewness of -0.33. The benchmark
Sharpe ratio is thus close, for example, to the value of 0.31 for the HML trade reported in Lustig et al.
(2014) for their set of developed countries, over a similar sample period and using equal weighting and
bid and ask quotes. When one currency is excluded from the carry trade on each trading day, practically
no change is observed, but when two currencies are excluded, the Sharpe ratio increases to 0.41, while
skewness drops to -0.57. When three to six currencies are excluded, the Sharpe ratios remain somewhat
higher than the benchmark ratio (between 0.41 and 0.46), with the differences not statistically significant.
However, skewness improves sharply in three out of these four cases and turns positive on two occasions
(and as high as 0.21 on one), whereby two of the associated p-values are below 5% and another one
equals 10%. These findings indicate a possible two-dimensional beneficial effect of excluding three or
more of the G-10 currencies, given that both the Sharpe ratio and skewness improve, albeit not always in
a statistically significant way. This effect is further confirmed by the enhanced trade that excludes seven
currencies: the Sharpe ratio is now 0.61, while skewness is positive, and both are marginally significantly
different from the benchmark values.
Our findings echo previous results, where significant improvements in skewness are obtained without
9
a substantial change in the Sharpe ratio (for example, the option-hedged carry trades in Burnside et al.
(2011). However, what is surprising in our case is that the improvement of the return profile is (i) in both
dimensions, and (ii) achieved simply by excluding currencies from the symmetric carry trade. Addition-
ally, the two-dimensional improvement is achieved by a procedure that maximizes the Sharpe ratios alone,
without considering the skewness of the returns so obtained.
C Identity of the excluded currencies
While on each trading date the enhanced trade re-considers the available return history and thus can
potentially deliver a different set of currencies to be excluded, we consistently observe the same currencies
to be excluded. Panel B of Table 2 shows the number of months that each G-10 currency is excluded by the
enhancement rule from Section A over the 234-month sample period of enhanced trading. In particular, it
shows how many times the respective currency is the first, or among the first two, or among the first three,
etc., to be excluded from the carry trade.
The consistency is observed most clearly with respect to the first three currencies excluded. Specifi-
cally, AUD is the first to be excluded on 135 out of the 234 trading dates in the sample. Furthermore, it
is among the first three currencies to be excluded on a total of 192 dates. Similarly, NOK is among the
first three excluded on 219 occasions, and JPY is among them on 214 occasions. These three currencies
appear to be by far the most detrimental to carry trade Sharpe ratios - no other currency is ever excluded
first, and only the EUR has been excluded second or third more than a handful of times.
The next currencies to be the most often excluded are the EUR, NZD, CAD and CHF, and while the
order of their exclusion is somewhat ambiguous, these are the obvious further candidates for exclusion
by the enhancement rule. The remaining three currencies are clearly found valuable by the rule: GBP
and SEK are among the first seven to be excluded only on about 60 occasions each, and in fact are never
among the first four excluded. Most conspicuously, however, the USD is never among even the first seven
10
excluded currencies, reminiscent of previous studies discussing the special role of the USD in the carry
trade (e.g., Lustig et al. (2014), Daniel et al. (2017)) from various perspectives.
These findings are surprising, as the enhancement rule consistently excludes from the carry trade pre-
cisely the prototypical carry trade currencies, like the JPY and AUD, which have been perpetually among
the lowest- or highest-yielding G-10 currencies, and feature commonly as examples in various carry trade
discussions. Because the consistency refers to the entire period since 1994, the recent financial crisis,
which witnessed drastic valuation changes in those currencies, cannot be solely responsible. Likewise, the
enhancement rule also tends to exclude the NZD and CHF, which have also been among the few highest-
or lowest-yielding currencies over our sample period.
The design of the enhancement procedure, as described in Section A, leaves open the possibility that
on some date no improvement of the Sharpe ratio can be achieved after certain number of exclusions,
whereby no further currencies are excluded and the respective enhanced trades are assigned the SC return
for the next trading period. This possibility is of some concern, as it could blur the distinction between
enhanced trades that exclude a different number of currencies. However, Panel B in Table 2 reveals that
this has never happened in our sample, as evidenced by the fact that the sum of the numbers in the first
row equals 234, the sum of those on the second row equals 234 × 2, and so on. Therefore, on each date in
the sample period the enhancement rule has identified seven currencies to be sequentially excluded, and
hence seven distinct enhanced trades to be implemented.
In sum, Table 2 shows that the enhancement rule consistently excludes the same few currencies from
the carry trade, among which are those epitomizing the essential concept underlying carry trades that low
(high) yield currencies should be sold (bought). The surprising evidence presented in Table 2 thus calls
for a re-consideration of this concept and/or its implementation.
11
D Good and bad carry trades from fixed subsets of the G-10 currencies
Prompted by the finding that the dynamic enhancement rule excludes the same currencies over and
over, we now examine carry trades constructed with fixed subsets of the G-10 currencies. While staying
close to the spirit of the enhanced trades, the fixed subsets allow for better comparison with previous carry
trade results, which are similarly obtained using fixed sets of currencies over fixed sample periods. Our
choice of the fixed subsets is informed by the order of exclusion implied by Panel B of Table 2, which
shows that (i) the three currencies that are the least often excluded by the enhancement rule are the GBP,
SEK and USD, (ii) the next three least often excluded are the CAD, NZD and CHF, whereas (iii) the AUD,
NOK and JPY are the most often excluded currencies.
In particular, we construct five carry trades from fixed subsets, which (i) exclude only the AUD, NOK
and JPY, (ii) include the GBP, SEK and USD, together with any of the three possible pairs from the CAD,
NZD and CHF, and (iii) keep only the GBP, SEK and USD. These carry trades are designed to illustrate
the properties of enhanced carry trades, and we denote them by G1 to G5, a notation we shall clarify
shortly. The first column of Table 3 displays the codes of the currencies included in each of these five
trades. We also consider the trades complementary to G1-G5, which include the currencies that are left
out of each of these trades, and denote these complements by B1 to B5, respectively, with currency codes
again displayed in the first column of Table 3. For example, only the three most often excluded currencies
(AUD, NOK and JPY) enter the B1 carry trade.
In addition, we consider a larger set of trades which can represent more broadly the enhanced carry
trades: it consists of 18 trades from five currencies each, and is denoted by GC, whereby each trade
includes the three least often excluded currencies (GBP, SEK and USD), together with any possible pair
from the remaining currencies which has none or only one of the three most often excluded currencies
(AUD, NOK and JPY). This choice yields a reasonably large cross section of trades which maintains the
12
predominant presence of currencies that are preferred by the enhancement rule. Again, we also consider
the 18 complementary trades, and denote them by BC. Despite creating many carry trades from only five
currencies, the average correlation among the returns of the 18 good trades is 0.66, and thus lower, for
example, than the average correlation among the 25 value-weighted Fama-French portfolios sorted on size
and book-to-market for the same period, which is 0.80.
Table 3 presents results for the SC trade, the G1-G5 and B1-B5 trades, and the GC and BC trades
described above, using the entire sample period from 12/1984 till 6/2014. Shown are annualized average
returns, return standard deviations and Sharpe ratios, as well as skewness. For the GC and BC trades we
show averages of these quantities. Also reported are p-values for tests of differences between the Sharpe
ratios and skewness coefficients, similar to those in Table 2. In the last two lines, the first (second) number
in parentheses shows how many of the 18 corresponding individual estimates for the GC or BC trades
are significant at the 5% (10%) level. Where p-values are (not) in square brackets, the null hypothesis is
that the Sharpe ratio or skewness of a G1-G5 trade or GC trade does not exceed that of the corresponding
B1-B5 trade or BC trade (SC trade). Note that over the full sample period the benchmark Sharpe ratio and
skewness remain close to those reported in Panel A of Table 2 for the shorter period since 1994.
The G1-G5 trades exhibit invariably higher average returns than the SC trade. In addition, their average
returns and return standard deviations tend to increase as the number of currencies in a trade decreases.
The Sharpe ratios of the G1-G5 trades all exceed the benchmark Sharpe ratio (in two cases by a factor of
about two), with the difference statistically significant at the 5% significance level in three cases out of
five. Skewness increases in three cases for the G1-G5 trades, even though this increase is significant only
for G1. Overall, these five trades reproduce the features that characterize the enhanced trades in Table 2.
In contrast, the complementary trades B1-B5 fare much worse. The average returns are often two to
three times lower than those of the SC trade, whereas the standard deviations are on average twice higher,
13
leading to much lower annualized Sharpe ratios, which are between 0.04 and 0.18. In addition, the return
skewness is markedly more negative for these complementary trades, averaging -0.77 (versus -0.11 for the
G1-G5 trades). Furthermore, the p-values shown in square brackets, pertaining to tests of the differences
in Sharpe ratios and skewness between the corresponding G1-G5 and B1-B5 trades are below 0.02 for
four out of five Sharpe ratios, and show three (one) rejections at the 5% (10%) level for skewness.
The relatively high Sharpe ratios and slightly negative or positive skewness of the G1 to G5 trades earn
them the label ”good” carry trades (”G” for good). Analogously, we refer to the B1 to B5 trades with low
Sharpe ratios and strongly negative skewness as ”bad” carry trades (”B” for bad), from now on.
Turning to the larger sets of GC and BC trades, each constructed from five currencies, the GC average
returns (Sharpe ratios) are on average three (three and a half) times higher than those for the BC ones, and
the GC skewness is on average twice lower (in absolute terms), whereby the differences are statistically
significant in about half the cases. The Sharpe ratios for the GC trades are significantly higher than those
for the SC trade in one third of the cases, in line with what was observed for the comparable G2 to G4
trades.
To further illustrate the properties of the GC and BC carry trades, Figure 2 plots their Sharpe ratios
versus skewness, similar to Figure 1, with unfilled circles and black dots, respectively. The distinction
is sharp and clear in the Sharpe ratio dimension, where, with no exception, the GC trades dominate the
BC trades, thus justifying their classification as good trades. On the other hand, a few GC (BC) trades
display low (relatively high) skewness, hence the distinction is not as clear in this dimension, even though
on average the skewness of the BC trades is still twice lower, consistent with the bad trades classification.3
In sum, eliminating some typical carry trade currencies, such as the AUD, JPY and NOK, from the
3In unreported results we find that when trades from five currencies involve the three least often excluded currencies (GBP,SEK, USD) in each case, but are now combined with any other pair that excludes the JPY, the good and corresponding badtrades deliver striking separation in both the Sharpe ratio and skewness dimension.
14
currency set leads to good carry trades, with Sharpe ratios and skewness mostly higher than those of the
SC trade, and the complementary bad carry trades that involve the typical carry trade currencies. Also,
with the exception of the G1 and B1 trades, the correlations between the various trades and SC are always
higher for the bad trades (on average 0.80, versus 0.67 for the good trades).
E Statistical significance of the distinction between good and bad carry trades
Table 3 shows that the distinction between good and bad carry trades is economically and statisti-
cally important. However, the statistical evidence must be interpreted with caution. In particular, the
reported p-values rely on the block bootstrap procedure under the alternative, developed by Ledoit and
Wolf (2008) (see Appendix OA-II). While this procedure accounts for certain finite-sample properties of
the distribution of currency returns, it does not reflect two aspects of our good carry trades. First, they are
constructed using information from the enhancement procedure, as reported in Panel B of Table 2, and
thus the procedure suffers from look-back bias. Second, the enhancement procedure applied to a finite
sample is bound to lead to improved Sharpe ratios, even if in population all 10 currencies are necessary to
attain optimal results. Therefore, a modified test is needed to assess the statistical contribution as fairly as
possible. We emphasize, however, that the results in Table 3 need not be statistically significant to impact
carry research: the finding that the prototypical carry trade currencies, if anything, worsen or certainly do
not provide a positive contribution to carry returns, suffices.
With respect to the look-back bias, starting the sample in 1994, rather than 1984 weakens the statistical
significance somewhat, but we still retain significance at the 10% level for the majority of the trades (not
reported). A full correction for this bias would require a much longer sample where we actually let the
procedure choose which currencies to exclude ex-ante, before we record trading results.
Incorporating the selection procedure into a test of statistical significance is harder, because it requires
creating a benchmark world in which carry trades still have realistic attractive returns, but somehow the
15
identity of the currencies contributing to these returns is randomized. Appendix OA-III describes in de-
tail a procedure creating entirely randomized individual currency returns which nonetheless reproduces
exactly the returns of standard carry (SC) in each randomized sample. We then apply our enhancement
strategy to 1000 such randomized samples, finding that the selection procedure biases the Sharpe ratios of
the good trades upwards by about 0.15 and that only the G5 (at the 5% level) and G3 (at the 10% level)
deliver statistically significant improvements in Sharpe ratios, using proper t-statistics.
Thus, there is no overall strong statistical evidence that the enhancement procedure delivers signifi-
cantly higher Sharpe ratios. However, it remains the case that the prototypical, ”skewed” carry currencies
can be removed from the trade without worsening performance.
IV Good carry trades as currency market risk factors
Lustig et al. (2011) suggest as a key currency market risk factor the return of a trading strategy
that each month goes long (short) a portfolio with the highest (lowest) forward differentials. This is
obviously a symmetric carry trade strategy and is denoted here as ”HMLFX ” (to be distinguished from the
Fama-French HML factor used in Section V). Creating test portfolios by ranking currencies on forward
differentials, they find that the covariation with HMLFX largely explains the difference in average returns
between these portfolios. Furthermore, they propose HMLFX as a proxy for a global risk factor in a no-
arbitrage model explaining the results, and show that it is also related to a measure of aggregate stock
market volatility. Menkhoff et al. (2012) conduct a similar exercise using a global exchange rate volatility
factor as a proxy for the global risk factor. We now revisit these findings by considering the good carry
trades as risk factors and comparing their performance with that of the previously used currency market
factors.
16
A Test assets and risk factors
The test assets in our pricing tests are five portfolios of currencies of developed countries (denoted
”Developed”), and six portfolios which also include emerging market currencies (denoted ”All”), created
by sorting the respective set of currencies on forward differentials, and taken from Adrien Verdelhan’s
website for the period ending in 12/2013. We consider these 11 portfolios together in our tests, and not
the ”All” and ”Developed” separately, as Lustig et al. (2011) do. The larger cross section poses a higher
hurdle to the various risk factors that are examined and compared.4 Our versions of the 11 portfolios
account for transaction costs.
Moving to the risk factors, first we use HMLFX (”All” version) as in Lustig et al. (2011) (and available
at Verdelhan’s website). Next, we use a mimicking portfolio for the innovations in foreign exchange
volatility (denoted ”FXVol”) as in Menkhoff et al. (2012). Finally, we also consider as risk factors the good
carry trades G1-G5 to contrast their performance, particularly with HMLFX . Because the correlations
between the G1-G5 trades and HMLFX (FXVol) are on average 0.39 (-0.29), and do not exceed 0.55
in magnitude multi-collinearity concerns do not arise. The respective correlations for the B1-B5 trades
average 0.64 (-0.71), suggesting a closer relation between the previously considered currency market
factors and our bad carry trades.
B Design of asset pricing tests
We adopt a standard asset pricing framework, following Cochrane (2005, Chapters 12 and 13), and
consider linear factor models, both in their beta representation and stochastic discount factor (SDF) form,
4While using some currencies twice in each test, the average correlation between the six ”All” portfolios and five ”Devel-oped” portfolios is only 0.74, which is just slightly higher than the average correlation among the ”Developed” (0.72) or the”All” portfolios (0.68). Moreover, we have verified that the relative performance of the risk factors separately on the ”All” and”Developed” portfolios remains largely the same.
17
assuming SDF’s specified as:
(3) mt+1 = 1 − b′ ( ft+1−E[ f ]).
In (3) ft+1 is a K× 1 vector of risk factors and b is a conformable constant vector of SDF coefficients.
Without loss of generality, we set E (mt+1) = 1, given that excess returns of test assets are used.
The SDF form of a pricing model is E[rxit+1mt+1] = 0, where rxi
t+1 are the excess percentage returns
of the test assets, indexed by i. The beta representation of the pricing model is E[rxit+1] = λ′ βi, with
systematic risk exposures for asset i given by the vector βi, and λ a vector of factor risk prices. The vectors
βi are estimated by GMM from time-series regressions of returns rxit+1 on the factors, and λ is estimated
from a cross-sectional regression (without a constant) of average returns on the β’s. We report the SDF
coefficients b and factor risk prices λ with corresponding p-values, as well as p-values for the χ2 statistic
testing if the pricing errors are jointly equal to zero (see, e.g., Cochrane (2005, page 237)).5
C Good carry trades in competition with other currency market risk factors
Table 4 shows the results of tests which compare the performance of HMLFX and the good carry trades
as risk factors. As in Lustig et al. (2011), each test also includes the dollar factor, denoted RX, which
is the average excess return of their basket of currencies held long against the USD. In each of the two
panels of the table, the first line refers to a model with the RX and HMLFX factors alone, the next five
lines to models with RX and each of the good carry trades G1-G5, and the remaining five lines to models
5Denoting by Rxt+1 the N×1 vector of the rxit+1’s, the moment conditions we use are:
g =
[E[Rxt+1mt+1]
E[ ft+1−E[ ft+1]]
]=
[E[Rxt+1−Rxt+1( f ′t+1−E[ f ′t+1])b]
E[ ft+1−E[ ft+1]]
].(4)
The weighting matrix defining which moments are set to zero is a =
[d 00 IK
], where d = E[Rxt+1 f ′t+1−Rxt+1E[ f ′t+1]].
Further, if µ = 1/T ∑Tt=1 ft and Rx = 1/T ∑
Tt=1 Rxt , where T is the length of the return time series, then the GMM estimates
of b are (d′d)−1d′Rx, and that of E[ ft+1] is µ. The standard errors of the b estimates are obtained from the covariance matrix
1/T (d′d)−1d′Sd(d′d)−1, where S is an estimator of ∑∞j=−∞ E[ut+1u′t+1− j] and ut+1 =
[Rxt+1mt+1
ft+1−µ
]. As in Lustig et al.
(2011), we use one Newey-West lag throughout to estimate S.
18
combining RX, HMLFX and each of the G1-G5 trades. The top panel summarizes results from time-series
regressions of each of the test assets, and reports average coefficient estimates and (in parentheses) the
number of respective estimates that are significant at the 5 or 10% confidence levels. (The regression
results for each individual test asset are shown in the Online Appendix, Table OA-6.) The bottom panel
reports both the prices of risk λ and the SDF coefficient estimates b. The latter are key in evaluating
the relative importance of alternative factors for pricing a given cross section (see, e.g., Cochrane (2005,
Chapter 13.4)).
The top panel of the table does not reveal important differences between HMLFX and the good carry
trades: the slope coefficients β in the time-series regressions are similarly significant; the R2 related to
HMLFX is slightly higher, but so are the respective intercepts α. When entering the regression jointly,
the two factors also show similar significance, with the HMLFX coefficients remaining negative on aver-
age, but the coefficients on the good trades turning all positive on average. The RX factor always has a
statistically significant slope coefficient of around 1.1. In the bottom panel of Table 4, all two-factor mod-
els (RX with either HMLFX or a good carry trade) show significant prices of risk λ for HMLFX and the
good trades (at the 5% level), but not for the RX factor. However, in the three-factor models the p-values
increase somewhat for HMLFX , and in three cases become significant only at the 10% level, while the
significance remains unaffected for the λ’s of the good trades.
An essential difference, however, is observed with respect to the SDF coefficients b. In the two-factor
models, the b-coefficient for HMLFX is significant at the 10% level only, but at the 5% level for all
good trades. However, in the three-factor models the b coefficients turn highly insignificant for HMLFX ,
whereas for the good carry trades they remain significant at the 5% level in three of the five cases, and at
the 10% level in one case. Moreover, the test for the pricing errors being jointly equal to zero rejects in
this sample for the two-factor model with RX and HMLFX with a p-value of zero, while the corresponding
19
p-values for the models with RX and a good trade are all above 0.10, except for G3 where the p-value is
0.09. The test fails to reject the three-factor models at the 5% level for all specifications. In addition, the
b coefficients appear similar across different specifications for the good trades, but not for the HMLFX
factor, where the sign switches across specifications. The results in the bottom panel of Table 4 clearly
favor the good carry trades over HMLFX as risk factors explaining the returns of the interest rate-sorted
currency portfolios.
Table OA-1 in the Online Appendix shows results from analogous tests, but with the currency volatility
factor FXVol replacing HMLFX . The conclusions remain robust: the good carry trades again win the horse
race, with p-values for all their SDF coefficients equal to 0.01 or lower, while these p-values are never
below 15% for FXVol.
D Return predictability of good and bad carry trades
The cross-sectional tests we have conducted follow the extant literature and assume constant prices of
risk and betas. It is surely conceivable that these assumptions are violated and thus that additional factors
may affect the unconditional cross section of currency returns (see e.g. Jagannathan and Wang (1996)).
There is, in fact, evidence of carry return predictability. Bakshi and Panayotov (2013) document that
commodity index returns and exchange rate volatility strongly predict carry trade returns. Further, Ready
et al. (2017) find time-series predictive ability of an index of shipping costs, the Baltic Dry Index (BDI),
for carry trade returns. They primarily investigate an unconditional carry strategy that is always long
the currencies of commodity exporters (commodity-producing countries) and short those of commodity
importers (countries producing final goods), which is a key component of their model.
In Table 5 we reconsider the evidence for time-series predictability from the perspective of good and
bad carry trades. To follow closely the empirical design in the two studies cited above, we use log returns
of equally-weighted good and bad carry trades. The commodity index predictor is defined as the three-
20
month log change in the CRB index. Exchange rate volatility, σavgt , is the cross-sectional average of
the annualized standard deviation of the daily log changes over each month t for each of the G-10 spot
exchange rates against the USD. The volatility predictor at the end of month t (and used to predict the
return for month t +1) is then the three-month log change ln(σ
avgt /σ
avgt−3). The shipping cost predictor is
the three-month log change in the BDI. As in Bakshi and Panayotov (2013), we show in-sample predictive
slope coefficients β and their p-values, using Hodrick (1992) standard errors, adjusted R2’s, and p-values
for the MSPE-adjusted statistic of Clark and West (2007).6 In addition, we show an (out-of-sample)
measure of the economic significance of the predictability, using the following strategy: if the predicted
carry return for month t + 1 is positive (negative), the strategy enters a carry trade at the end of month t
(no position is taken and the strategy’s return for month t + 1 is zero). The reported measure ”∆ SR” of
economic significance equals the difference between the Sharpe ratio of the trading strategy, implemented
with the respective subset of G-10 currencies, and the corresponding carry trade as shown in the first
column. The predictive regressions use an expanding window with initial length of 120 months.
Table 5 shows clear differences between the return predictability results for good and bad carry trades.
Out of 15 possible combinations with the three predictors, the G1-G5 trades show a significant predictive
slope on three occasions, whereas the B1-B5 trades record 13 occasions with p-values not higher than 0.05
and another one with a p-value below 0.10. The average predictive R2 is 0.7% for the G1-G5 trades and
2.2% for the B1-B5 trades. The MSPE statistics show significant out-of-sample predictability in one case
(out of 15) for the G1-G5 trades and in 13 cases for the B1-B5 trades. Finally, exploiting the predictability
in dynamic trading does not materially impact the Sharpe ratio for the G1-G5 trades (the average change
is -0.005), while it mostly improves the Sharpe ratio for the B1-B5 trades, on average by 0.10. The
6MSPE stands for ”mean squared prediction error”. The statistic is obtained using ft+1 = (yt+1−µt+1)2− [(yt+1− µt+1)
2−(µt+1− µt+1)
2], where µt+1 is the prediction for month t +1 from a predictive regression yt+1 = a+bxt + εt+1, and µt+1 is thehistorical average of y. Both µt+1 and µt+1 are estimated using data up to month t. The null hypothesis is that µt+1 does notimprove on the forecast which uses µt+1 as the predictor. The test statistic is the t-statistic from the regression of ft+1 on aconstant, for which we report one-sided p-values.
21
improvement is economically large , because the Sharpe ratios for bad carry trades are often below 0.10
(see Table 3). The above patterns are confirmed by the results from the GC and BC trades, where again
the GC trades show insignificant predictive slopes for two of the predictors, twice smaller predictive R2’s,
rarely significant out-of-sample predictability, and on average a reduction in Sharpe ratios from exploiting
predictability by 0.01, in contrast to the BC trades which exhibit an increase by 0.09 in Sharpe ratios on
average.
Our predictability results echo some findings in Ready et al. (2017). The CRB commodity index most
strongly predicts the returns of the factor that they denote IMX, which is long AUD, NZD and NOK, and
short JPY and CHF, as often true for our bad trades.7 Therefore, our Table 5 confirms the predictive ability
of the CRB and BDI for a ”commodity focused” carry trade as implied by their commodity trade model.
Our contribution here, however, is to highlight the similarity between the commodity-based trade and our
bad carry trades, and the fact that a commodity-based interpretation of carry trade returns reflects mostly
features of bad carry trades.
Our results therefore qualify the prevailing carry return predictability story. A carry trade that focuses
on the prototypical carry currencies is rather unattractive, but its return properties can be enhanced by ex-
ploiting return predictability. In contrast, our good carry trades have attractive properties which, however,
cannot be enhanced by the predictors previously identified in Bakshi and Panayotov (2013). It remains, of
course, conceivable they are predictable by other variables.
V Good and bad carry trades and previous carry interpretations
This section reconsiders previous studies of carry trades from the good-bad carry trade perspective.7They also show that a complement to the IMX trade (denoted CHML) is not predictable at all by the CRB or BDI, and also
has practically zero skewness, similar to some of our good trades. However, the Sharpe ratio of CHML is still below that of theirversion of the standard carry trade (0.85 vs. 0.95 in their sample and without transaction costs), hence their orthogonalizationprocedure fails to identify a good carry trade.
22
A Explaining carry trade returns with equity market risk factors
We start by re-examining a key result in the literature stating that standard (linear) equity market
factor models cannot explain the time variation in carry returns, which appear uncorrelated with these
risk factors in normal times, but correlate highly with them in crisis times (e.g., Melvin and Taylor (2009),
Christiansen, Ranaldo, and Soderlind (2011)). We examine three models: (i) the Fama-French three-factor
model, following Burnside et al. (2011), (ii) a three-factor model with the market factor, the global equity
volatility factor used in Lustig et al. (2011), and their product, and (iii) a model with two factors which
explicitly distinguish the down- and up-moves of the equity market, in the spirit of Lettau, Maggiori,
and Weber (2015). The latter two models effectively exhibit a non-linearity that may capture the time-
variation in the correlation mentioned above. To conserve space, we relegate detailed results to the Online
Appendix, summarizing the key results here.
Let’s start with the model featuring a market factor (denoted MKT), proxied by the total return of
the MSCI-World equity index, in excess of the risk-free rate and expressed in USD, an equity volatility
factor (EqVol) constructed as in Lustig et al. (2011), and the interaction term (the product of MKT and
EqVol). Table OA-2 shows that in time-series regressions of carry trade returns on the three risk factors
the main difference between good and bad trades is in their loadings on the product factor. These are
typically negative, albeit rarely significant, for the good trades, while they are positive, much larger in
magnitude, and almost always significant at the 5% significance level for the bad trades. Given that
increases in volatility tend to coincide with market downturns, the market exposure of the bad trades
increases substantially in bad times, making them under-perform in times of crisis.
We also perform GMM-based cross-sectional tests on the GC and BC return cross sections. For the
GC trades, the risk price for the MKT factor is significant at the 5% level, while for the BC trades no risk
price is significant, although the model is not rejected for either of the two cross sections. When we run a
23
simple OLS regression of actual average returns on a constant and the model-based expected returns, we
obtain an R2 of 0.67 for the GC trades, and 0.29 for the BC trades. The combined evidence suggests that
this three-factor model does not adequately describe the returns of the bad carry trades, but still saliently
reveals the high exposure of these trades to the equity market during high-volatility periods. In contrast, a
significant price of risk for the market factor and tighter link between model expected returns and average
returns show the promise of the model to provide a risk-based interpretation of good carry trades.
The Online Appendix further shows quite similar results for the model with an Up- and Down-market
factors. Table OA-3 shows that good (bad) carry trades load primarily on the Up (Down)-market factor,
with beta exposures being economically and statistically very different across the two types of trades.
In the cross-sectional tests, the prices of risk for both factors are significant; the pricing errors are not
statistically different from zero and the model generates expected returns highly (weakly) correlated with
good (bad) carry trades.
The Online Appendix and Table OA-4 report analogous results for the Fama-French three-factor
model. Here the time-series regressions reveal that good carry trades do not load much on any of the
three factors, and retain significant alphas relative to the model. In contrast, the bad carry trades feature
significantly higher regression slope coefficients on all three factors and it is striking that their SMB and
HML exposures are positive and economically meaningful (often even above 0.10). However, the Fama-
French model fails to fit expected returns cross-sectionally, with all prices of risk being insignificantly
different from zero for both good and bad carry trades.
In sum, the evidence from Tables OA-2 to OA-4 provides (weak) support for the ability of risk fac-
tors from the equity market to explain the returns of the good carry trades. Our results are not directly
comparable to studies analysing numeraire-dependent carry trades, such as Daniel et al. (2017).
24
B Currency crashes as an explanation for the carry return puzzle
One established explanation for the carry trade’s profitability is that it reflects compensation for the
negative return skewness or crash risk, inherent to these trades. For example, Brunnermeier et al. (2009)
argue that ”investment currencies are subject to crash risk, that is, positive interest rate differentials are
associated with negative conditional skewness of exchange rate movements.... The skewness cannot easily
be diversified away, suggesting that currency crashes are correlated across different countries .... This
correlation could be driven by exposure to common, crash-risk factors”. If agents exhibit a preference for
positive skewness, an equilibrium model may generate negatively skewed returns and high Sharpe ratios
for the carry trade.
However, the crash risk hypothesis is not consistent with our findings from good and bad carry trades
(see Figure 1 and Table 3): good carry trades have relatively high Sharpe ratios and slightly negative
(or even positive) skewness. The assertion in Brunnermeier et al. (2009) that the negative skewness in
carry trade returns cannot be diversified away must also be qualified. We have demonstrated that, in fact,
skewness can be dramatically improved by judiciously removing currencies from the carry trade, without
impairing profitability. Studies relying on option market data (e.g., Burnside et al. (2011), Jurek (2014))
have criticized the crash-risk hypothesis before, because options can essentially hedge away the crash risk
without undermining much the carry trade’s profitability.
VI Further exploration of good and bad carry trades
In this section we embark on a more detailed examination of the good and bad carry trades, trying to
set the stage for future work that will hopefully clarify fully the economic interpretation of our findings.
First, we reflect on the return components of various carry trades and how they contribute to the differential
performance of the good and bad trades. Second, the good carry trades always include the USD (it is never
25
excluded in our selection procedure). There is a burgeoning literature stressing the special nature of the
USD in international financial markets: Adrian, Etula, and Shin (2015) associate increased global dollar
funding with expected currency depreciations; Hassan (2013) argues that economies representing a larger
share of world wealth have low interest rates and low risk premiums, whereas Maggiori (2013) ascribes
a low premium to holding the USD to its role as a reserve currency. Lustig et al. (2014) explore a new
trade, denoted ”dollar carry”, which goes long (short) in all foreign currencies against the USD with equal
weights when their average interest rate differential relative to the USD is positive (negative). The dollar
carry trade has a very attractive Sharpe ratio, substantially higher than that of SC, raising the issue that
we may have simply repackaged dollar carry into our good carry trades. We show that this is not the case,
and these two types of trades, while correlated, are economically distinct.
A The sources of good and bad carry returns
In Table 6, we decompose carry trade returns into an interest rate (or forward differential) component,
and an exchange rate change component. SC derives more than 100% of its returns from the interest rate
component, i.e., the investment currencies do depreciate and/or the funding currencies do appreciate, but
the exchange rate component is sufficiently small relative to the ”carry” to leave an attractive return on the
table. Bad carry trades have higher carry return components, both in absolute terms (and the difference is
statistically significant) and in relative terms, but even more negative exchange rate components, so that
lower returns than those for standard carry are obtained. In contrast, good carry trades derive their returns
both from the carry and exchange rate components. Their carry component is on average about 20 basis
points lower than that of SC in three cases (and statistically significant at the 5% level for the G4 trade),
while it is significantly higher for the G3 trade (even if still lower than the carry of any bad trade).
The contrast between the carry contributions to the returns of bad and good trades is illustrated in
Figure 3. The graph plots total average return on the horizontal axis, and the ratio of carry to total return
26
on the vertical axis for all trades considered in this paper (18 GC trades, 18 BC trades, as well as the G1
and G5, and B1 and B5 trades). The graph also includes the standard and dollar carry trades. Bad carry
trades have lower returns and much higher carry-to-return ratios; good carry trades have higher returns,
and derive between 50 and 100% of their returns from carry (the G5 trade being the only exception).
These results suggest that the unbiasedness hypothesis may not be strongly rejected for bad carry
currencies, which include the prototypical carry currencies. Recall that a necessary condition for a carry
trade to deliver excess returns is that the unbiasedness hypothesis does not hold, at least for some period
of time (see Bekaert, Wei, and Xing (2007) for recent tests of the hypothesis). However, when examining
standard regressions testing the unbiasedness hypothesis for the four pairs containing prototypical carry
currencies, AUD/JPY, NZD/JPY, AUD/CHF and NZD/CHF (see Table 7), we find no strong rejections
of the hypothesis. In particular, we regress future exchange rate changes onto a constant and the current
forward differential, and the null hypothesis is that the constant is zero and the slope coefficient is one.
The constants (slope coefficients) in all four regressions are insignificantly different from zero (one).
Most saliently, the slope coefficient for the AUD/JPY regression is 0.92, and thus remarkably close to one.
However, our analysis reveals the NOK also to be a ”bad” currency, more so than the CHF and the NZD.
Interestingly, Table 7 shows that the slope coefficient in unbiasedness regressions of the NOK relative
to the CHF and JPY is either not significantly different from one, or exceeds one by a large amount,
indicating an expected depreciation of the NOK relative to the JPY when the NOK interest rate exceeds
the JPY one. This is partially counteracted by a positive and significant constant.
The decomposition and the regression results above also suggest that the good carry trades are likely to
be more ”active” than the bad or standard carry trades, i.e., they likely involve more frequent re-balancing.
The insightful paper by Hassan and Mano (2015) decomposes the carry trade into a ”static” trade (which
goes long (short) currencies with unconditionally low (high) forward differentials) and a dynamic trade,
27
which also helps explain deviations from unbiasedness. Such deviations, driven by the slope coefficient
in the unbiasedness regressions being different from one, lead to dynamic trades when forward premiums
are high or low relative to their unconditional means. However, carry trades can also be profitable simply
through non-zero constants in the unbiasedness regressions (see Bekaert and Hodrick (2012, Chapter 7)).
Table 8 provides some evidence on the dynamic nature of the various carry trades. We create a dummy
variable that records the proportion of currencies that change position (from long to short or vice versa) at
each point of time. For example, for a completely static trade this proportion is zero, whereas a trade where
half the currencies switch positions at each point of time would record 50% on this measure. Furthermore,
since the dynamic nature may be related to the number of currencies in the trade, it is important to take
sampling error into account. The table shows the sample averages of these proportions, together with 95%
confidence intervals, computed using the bootstrapped carry trade returns employed in Section E.
Clearly, the good carry trades are more ”dynamic” than the bad trades and the SC trade. Note that the
average proportions for good trades are invariably above the confidence interval for SC (and vice versa for
the bad trades). Yet, the proportions for the good carries and SC are relatively highly correlated in the time
series (ranging between 49% and 82%), suggesting that these trades switch currency positions at roughly
similar times. For bad carry trades the proportion of switches is typically within the confidence interval of
SC, except for the B3 trade.
The last three columns in Table 8 report the ratios between the average returns of various static carry
trades (which are never re-balanced), and the returns of the corresponding good or bad carry trade, mim-
icking the Hassan and Mano (2015) methodology.8 The trades are constructed with all G-10 currencies,
as well as with the currencies entering the G1-G5 and B1-B5 trades. The various ”static trades” use as
weights the average forward differentials over the 12/1984-12/1994 period (the first 120 months of our
8We use log returns and currency weights equal to the demeaned and normalized forward differentials at each trading dateover 1/1995-6/2014.
28
sample), demeaned and normalized to have absolute values that sum to one. The weights are kept fixed
for the entire sample period 1/1995-6/2014, without ever re-balancing.
Hassan and Mano (2015) find that static trade returns account for about 70% of carry trade returns
(but the standard error on that estimate is substantial), whereas according to Lustig et al. (2011) this
proportion is between one third and one half. Analogously, the average return of the static SC trade in
Table 8 is about half of that of the original SC trade. Importantly, there is a clear distinction between the
relative performance of the static versions of the good and bad carry trades, with the ratios between the
corresponding average returns never exceeding 0.30 (and sometimes going negative) for the good trades,
but ranging between 0.60 and 1.2 for the bad trades. The distinction is even clearer in terms of Sharpe
ratios (see the last two columns), which for the good static trades rarely exceed 0.15, much worse than
their re-balanced counterparts. In contrast, the Sharpe ratios of the re-balanced and static ”bad” trades are
close to one another.
Hence, good carry represents a dimension of standard carry that is not well explained by its static
component. This is intuitive, because good carry trades tend to exclude currencies with either the highest
or lowest forward differentials, and thus do not have stable short and long positions. In contrast, and as
shown above, the currencies involved in bad carry trades typically switch less often from long to short
positions and vice versa. Our results thus confirm the Hassan and Mano (2015) decomposition for ”bad
carry trades”, but not for ”good carry trades”. Hassan and Mano (2015) split up carry trades in the
static carry trade we studied above and a ”dynamic” trade, which essentially exploits time-variation in
the relative ranking of currencies in terms of their forward differentials, relative to their unconditional
counterparts. This dynamic trade must necessarily be relatively more important for good trades, which
feature currencies with less extreme interest rate differentials relative to the dollar, and for which the
unbiasedness hypothesis does not hold. The dynamic trade therefore also contributes positively to the
29
trade exploiting deviations from unbiasedness (what they called the ”forward premium trade”). Do note
that our results are not entirely comparable to Hassan and Mano (2015) because they do not impose
symmetry on their carry trade, while we do.
B Good carry versus dollar carry
In this section, we characterize the differences and similarities between the dollar carry trade and our
good carry trades. First, note that dollar carry (hereafter DC for short) does not satisfy the standard condi-
tions for a carry trade as discussed in Section II. Carry trades go long (short) high (low) yield currencies,
whereas DC combines high and low yield currencies on one side of the trade. Going back to Table 6, the
last column reports the carry (i.e., interest rate) and exchange rate change components for DC, and the last
row of the table reports p-values for a test of equality between the carry components of the trade in the re-
spective column and DC. The DC trade derives most of its substantial returns from currency appreciation,
and only 22% from interest rate differentials. This proportion is significantly lower than that of any other
carry trade. Perhaps not surprisingly, the G5 trade, only featuring three currencies which include the USD
comes closest to DC. In Figure 3, the DC trade also represents somewhat of an outlier. Furthermore, when
constructing versions of DC from ”good” and ”bad” currencies separately, we find that their Sharpe ratios
are very similar.
Second, DC is much less dynamic than the good trades: the last row of Table 8 shows that it switches
positions more rarely (despite any switch involving all currencies). The switching proportions are also not
very correlated with those for the good trades (at most 39%) or the SC trade (47%). Furthermore, Table 8
(column ”days w/o switch”) shows that while the typical proportion of days when no currency switches
position from long to short or vice versa ranges between 0.60 and 0.80 in carry trades, it is 0.93 for DC.
Third, because good carry trades eliminate some non-dollar exposure, they should be more correlated
with DC than SC or the bad carries are. Table 9 confirms this intuition, showing in Panel A that DC has
30
the highest correlation with good trades. However, for the G1, G2 and G3 trades the correlation is less
than 50%, and it does not exceed 70% for the G5 trade. Moreover, the correlations between the good
trades (except G5) and SC are higher than those for DC. Good carries thus preserve their close link with
the SC trade, and remain distinct from DC.
In Panel B of Table 9 we report on regressions which have the returns of SC, DC or good carries either
as dependent or independent variables. Both SC and good carries have explanatory power for the DC
trade, but SC is insignificant in two specifications. The R2’s are relatively low, being less than 40% in
all but one case. The DC trade mostly delivers significant alphas relative to these two factors, which is
not surprising, given its very attractive return profile. Next, both SC and DC have explanatory power for
good carry trades, with almost all slope coefficients featuring p-values below 1% and R2’s ranging from
43 to 75%. The coefficients on SC, however, are typically much larger than the coefficients on DC. The
G2, G3 and G5 trades still show significant alphas with respect to these two factors. Finally, the return of
SC is explained with R2s between 18 and 75% and all intercepts are insignificantly different from zero.
The explanatory power in this case comes predominantly from the good carries, as evidenced by the small
coefficients and some high p-values on DC.
Table 9 shows that neither good carries, nor DC can be perfectly spanned by other trades, despite
being correlated with them. In contrast, SC is spanned, and this is mostly due to the good carry trades.
Being proper carry trades, the good carries should therefore be viewed, unlike DC, as better versions
of SC.9 We also investigate the explanatory power of three economic factors for various carry trades,
including DC, namely, the global equity market volatility, global industrial production growth, proxied by
the OECD total growth10, and the residual from regressing the US industrial production growth onto the
9For further validation of this claim, in unreported results we consider creating a mean-variance efficient portfolio from DC,SC and one of the G1-G5 portfolios. When we do so, the good carries invariably get a large positive weight, always exceedingthat of DC, sometimes by factor of two or three, and SC is always shorted (except if paired with the G5 trade, when its weightis close to zero). In other words, good carries dominate DC and SC is pushed out.
10From stats.oecd.org, Monthly economic indicators, Production of total industry excluding construction, growth rate over
31
global growth variable (see Online Appendix Table OA-5 for the results). We find that, the various carry
trades are similarly related to the macro factors considered, but that DC appears to have no significant link
to these factors. Hence, an economic interpretation of the carry trade returns remains elusive.
Finally, DC and good carry trades both exhibit little skewness, but use very different mechanisms to
eliminate the impact of bad carry currencies in this respect. While the good trades simply remove the
currencies, DC puts such naturally ”long” and ”short” currencies on the same side of the trade. To see
this more clearly, consider the small table underneath, which shows average forward differentials against
the USD for the remaining G-10 currencies over our full sample period 12/1984 to 6/2014 (together with
the first three moments of the percentage returns of long positions in each currency against the USD, not
adjusted for transaction costs). All numbers are annualized and in percent, except for skewness. The bad
carry currencies (JPY, AUD and NOK) do not only have among the highest forward differentials, they also
are the most skewed. The good carry trades essentially remove these currencies and thereby do not worsen
and mostly improve the return-risk properties of the trade. Therefore, this skewness must be idiosyncratic
and not priced, or it must be endogenously generated by carry traders. Why this is the case remains an
important open question for further research, but it surely undermines any explanation of attractive carry
returns based on priced ”crash” risk.
C Revisiting the factor pricing of currency returns
We now compare the performance of DC and good carries as pricing factors for the interest rate-sorted
portfolios, discussed in Section C above. Table 10 reports results from pricing tests, which juxtapose DC
the previous month, seasonally adjusted
32
with the G1-G5 trades, but do not include the RX factor (which only shorts the USD and has correlation
of 0.53 with DC).11 The top panel summarizes, as in previous tables, results from time-series regressions
on individual portfolios, while the bottom panel presents results from cross-sectional tests. (The first-pass
regression results for each individual test asset are shown in the Online Appendix, Table OA-7.)
DC alone explains reasonably well the time-series behavior of the test assets - none of the intercepts
and all slope coefficients are significant, even at the 5% confidence level, with a relatively high R2 of 21%.
The performance of the good trades alone is similar with respect to the intercepts, while the slopes are
not always significant and the R2’s are much lower in three out of five cases. Moving to the cross-section,
however, we observe that the price of risk for DC is significant only with a p-value of 0.07, while three of
the good trades show significance at the 5% level. In addition, the test for the pricing errors being jointly
equal to zero rejects with a p-value of 0.03 for DC, but never rejects for individual good carries, even at
the 10% confidence level.
We also perform cross-sectional tests which include both DC and a good trade, as reported at the
bottom of Table 10. The price of risk λ is now statistically significant for DC at the 5% level in only two out
of five cases, whereas four out of the five p-values for the good carries are at 2% or below. The p-values for
the SDF coefficients b, which provide the proper horse race test, are all above 0.20 for DC. In contrast, four
of these p-values for the good trades are below or equal to 0.10. The relatively high correlation between
DC and the G5 trade may be the source of insignificant coefficients in this specification. Also note that
the b-coefficients for the good trades are quite stable (see also Table 4), whereas the b-coefficients for DC
switch sign across specifications. While the statistical significance in favor of good trades is borderline,
the conjecture that good trades may be simply reflecting features of DC is thus not supported here.11DC is short the dollar about 70% of the time in our sample, and its profitability is entirely driven by these short dollar
positions. Note that this is not true for the good carry trades, which gain both when the dollar is short and long.
33
VII Conclusion
This paper introduces ”good” and ”bad” carry trades, which are all constructed from subsets of the
G-10 currencies, but exhibit markedly different return properties, in terms of Sharpe ratios and skewness.
Surprisingly, trades that just exclude some of the typical carry trade currencies do perform better than
the benchmark SC trade, while trades that only include the typical carry currencies have inferior return
profiles. These findings challenge the conventional wisdom on the construction of carry trades from an
investor’s view point. Furthermore, the trades from subsets also challenge some of the available conceptual
interpretations of the carry trade. We document that several of these interpretations appear to be mostly
consistent with the bad carry trades, but are less applicable to good trades.
We find that good carry trades can serve as risk factors, able to explain a cross section of currency
portfolio returns, and in this role can drive out previously suggested risk factors, such as the HMLFX
factor of Lustig et al. (2011). Further, the returns of good carry trades can be explained to a certain extent
with risk factors from the global equity market. While good carry trades are more strongly correlated
with the ”dollar carry” trade of Lustig et al. (2014) than is the standard carry trade, good trades remain
symmetric carry trades, deriving the bulk of their returns from carry (i.e., interest rate differentials), and
offer a distinct return profile.
The results in this paper, even though largely focused on the statistical properties of carry trade returns,
should impact the study of carry trades in various directions. First, exploring crash risk or differentiating
fundamental risks of commodity producers versus exporters are unlikely fruitful avenues of research.
Second, our reported asset pricing tests can inform further risk-based interpretations of carry trade returns.
Finally, it can be promising to explore, in the spirit of Koijen, Moskowitz, Pedersen, and Vrugt (2015),
the notion of good and bad carry trades from financial assets other than currencies.
34
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37
Tabl
e1
Som
ere
cent
appr
oach
esto
carr
ytr
ade
cons
truc
tion
Thi
sta
ble
sum
mar
izes
thre
eas
pect
sof
carr
ytr
ade
cons
truc
tions
,as
adop
ted
inre
cent
stud
ies.
Itsh
ows
whi
chcu
rren
cies
are
empl
oyed
inth
etr
ade,
are
curr
enci
esgi
ven
equa
lwei
ghts
(pos
sibl
yam
ong
othe
rw
eigh
ting
sche
mes
),an
dw
heth
erth
eto
talw
eigh
tsof
the
long
and
shor
tsid
esof
the
trad
ear
eeq
ual.
The
botto
mpa
rtof
the
tabl
epr
ovid
essi
mila
rdet
ails
onse
vera
linv
esta
ble
carr
ytr
ade
inde
xes
(sou
rce:
FXW
eek
(Feb
,200
8)).
Whi
chcu
rren
cies
?W
eigh
tseq
ual?
Lon
gan
dsh
orte
qual
?
Bru
nner
mei
eret
al.(
2009
)G
-10
yes
yes
Cla
rida
etal
.(20
09)
G-1
0ye
sye
sJo
rda
and
Tayl
or(2
009)
G-1
0ex
SEK
yes
noA
ngan
dC
hen
(201
0)G
-10
plus
13ot
her
yes
yes
Bur
nsid
eet
al.(
2011
)20
deve
lope
dye
sno
Lus
tiget
al.(
2011
)L
ustig
etal
.(20
14)
Men
khof
feta
l.(2
012)
Rea
dyet
al.(
2013
)
nine
to34
,pl
usa
smal
ler
subs
etof
15de
velo
ped
yes
yes
yes
yes
yes
no yes
yes
Bak
shia
ndPa
nayo
tov
(201
3)G
-10
yes
yes
Has
san
and
Man
o(2
015)
nine
to39
,plu
sa
smal
ler
no(d
evia
tions
offo
rwar
dye
ssu
bset
of15
deve
lope
ddi
ffer
entia
lsfr
omth
eirm
ean)
Dan
iele
tal.
(201
7)G
-10
yes
and
no(v
ario
usw
eigh
ting
sche
mes
)ye
san
dno
Jure
k(2
014)
G-1
0ye
san
dno
(var
ious
wei
ghtin
gsc
hem
es)
yes
and
noB
arro
soan
dSa
nta-
Cla
ra(2
014)
27de
velo
ped
no(v
ol.s
cale
dde
viat
ions
offo
rwar
dye
sdi
ffer
entia
lsfr
omth
eirm
ean)
Som
ein
vest
able
carr
ytr
ade
inde
xes:
Bar
clay
sC
apita
l:In
telli
gent
G-1
0no
(por
tfol
ioop
timiz
atio
n)ye
sC
arry
Inde
xC
itigr
oup:
Bet
a1ra
nge
G-1
0ye
s(u
ses
9or
13m
ost
yes
liqui
dtr
adea
ble
pair
s)C
redi
tSui
sse:
Rol
ling
G-1
0an
dG
-18
no(p
ortf
olio
optim
izat
ion)
yes
Opt
imis
edC
arry
Indi
ces
Deu
tsch
eB
ank:
G-1
0H
arve
stIn
dex
G-1
0ye
s(t
hree
high
est-
and
low
est-
yiel
ding
)ye
sJP
Mor
gan:
Inco
meF
XG
-10
yes
and
no(f
ourp
airs
sele
cted
each
yes
mon
thto
optim
ize
the
risk
-ret
urn
ratio
)
38
Tabl
e2
Car
rytr
ades
cons
truc
ted
byse
quen
tially
excl
udin
gG
-10
curr
enci
esU
sing
mid
-quo
tes
fors
pot(
S t)a
ndon
e-m
onth
forw
ard
(Ft)
exch
ange
rate
sof
the
G-1
0cu
rren
cies
agai
nstt
heU
SD(E
uro
splic
edw
ithD
EM
befo
re19
99)
from
Dat
astr
eam
,we
calc
ulat
efo
rwar
ddi
ffer
entia
lsas
F t/S
t−
1.W
ithth
ese
we
cons
truc
tcar
rytr
ades
eith
erus
ing
allG
-10
curr
enci
es,o
rex
clud
ing
one
upto
seve
nof
thes
ecu
rren
cies
,as
expl
aine
din
Sect
ion
A.
Pane
lAre
port
sth
eira
vera
ges
(den
oted
”avg
.re
t.”,a
nnua
lized
anin
perc
ent)
and
SR’s
(bot
han
nual
ized
),as
wel
las
skew
ness
(den
oted
”ske
w”)
.T
hefir
stco
lum
nof
the
tabl
esh
ows
how
man
ycu
rren
cies
have
been
excl
uded
.T
heco
lum
nsde
note
d”p
-val
”ar
eob
tain
edus
ing
boot
stra
pco
nfide
nce
inte
rval
sfo
rthe
hypo
thes
isth
atth
ere
spec
tive
SRor
skew
ness
does
note
xcee
dth
ebe
nchm
ark
one
(see
App
endi
xO
A-I
I).T
hesa
mpl
epe
riod
is12
/198
4to
6/20
14(3
54m
onth
s),a
ndth
ein
itial
win
dow
for
calc
ulat
ing
SR’s
that
are
used
tode
cide
whi
chcu
rren
cies
shou
ldbe
excl
uded
is12
0m
onth
s.E
ach
colu
mn
inPa
nelB
show
sho
wm
any
mon
ths,
outo
fth
e23
4m
onth
sin
whi
chw
ese
arch
for
the
high
estS
R’s
,was
acu
rren
cyw
ithco
dedi
spla
yed
inth
efir
stro
w,t
hefir
stto
beex
clud
ed(r
owst
artin
gw
ith”1
”),o
ram
ong
the
first
two
excl
uded
(row
star
ting
with
”2”)
,etc
.
A.A
vera
gere
turn
s,Sh
arpe
ratio
s,sk
ewne
ssB
.Fre
quen
cyan
dor
dero
fexc
lusi
on
avg.
ret.
SRp-
val
skew
p-va
lN
ZD
AU
DN
OK
GB
PSE
KC
AD
USD
EU
RC
HF
JPY
SC1.
000.
32-0
.33
11.
050.
320.
51-0
.26
0.24
013
550
00
00
00
492
1.23
0.41
0.22
-0.5
70.
870
184
219
00
00
150
503
1.25
0.41
0.25
0.01
0.03
019
221
90
00
074
321
44
1.34
0.46
0.21
0.21
0.01
1422
423
40
082
015
210
220
51.
560.
400.
32-0
.01
0.10
106
224
234
041
820
230
1923
46
1.47
0.44
0.29
-0.3
50.
5317
123
423
436
5817
50
230
3223
47
3.17
0.61
0.10
0.04
0.10
172
234
234
6358
175
023
423
423
4
39
T abl
e3
Car
rytr
ades
cons
truc
ted
with
fixed
subs
etso
fthe
G-1
0cu
rren
cies
Thi
sta
ble
show
sav
erag
es(d
enot
ed”a
vg.
ret.”
,in
perc
ent)
,sta
ndar
dde
viat
ions
(”st
.dev
.”,i
npe
rcen
t)an
dSh
arpe
ratio
s(”
SR”)
,all
annu
aliz
ed,a
sw
ell
assk
ewne
ss(d
enot
ed”s
kew
”)fo
rth
em
onth
lyex
cess
retu
rns
ofse
vera
lca
rry
trad
es,
all
with
equa
lw
eigh
ts.
”SC
”de
note
sth
est
anda
rdca
rry
trad
eco
nstr
ucte
dw
ithal
lG-1
0cu
rren
cies
.The
G1
toG
5tr
ades
are
cons
truc
ted
usin
gth
ecu
rren
cies
with
the
disp
laye
dco
des,
and
repr
esen
tcom
bina
tions
ofcu
rren
cies
that
are
less
ofte
nex
clud
edby
the
enha
ncem
entr
ule
disc
usse
din
Sect
ion
Aan
dTa
ble
2.B
1to
B5
are
the
com
plem
enta
rytr
ades
,eac
hus
ing
the
curr
enci
esth
atha
vebe
enle
ftou
tof
one
ofth
eG
1-G
5tr
ades
.”G
C”
deno
tes
the
seto
f18
carr
ytr
ades
,eac
hco
nstr
ucte
dfr
omth
eth
ree
leas
toft
enex
clud
edcu
rren
cies
(USD
,GB
Pan
dSE
K),
com
bine
dw
ithan
ypo
ssib
lepa
irof
the
rem
aini
ngG
-10
curr
enci
esth
atco
ntai
nsno
neor
only
one
ofth
eth
ree
mos
toft
enex
clud
edcu
rren
cies
(AU
D,N
OK
and
JPY
).”B
C”
deno
tes
the
seto
f18
com
plem
enta
ryca
rry
trad
es,e
ach
usin
gth
efiv
ecu
rren
cies
left
outo
fone
ofth
e18
trad
esin
GC
.The
row
sco
rres
pond
ing
toth
eG
Can
dB
Ctr
ades
show
aver
ages
ofth
eav
erag
ere
turn
s,st
anda
rdde
viat
ions
,Sha
rpe
ratio
san
dsk
ewne
ssac
ross
the
resp
ectiv
e18
carr
ytr
ades
.T
heco
lum
nsde
note
d”p
-val
”sh
owp-
valu
esob
tain
edus
ing
boot
stra
pco
nfide
nce
inte
rval
s(s
eeA
ppen
dix
OA
-II)
.The
first
(sec
ond)
num
ber
inpa
rent
hese
ssh
ows
how
man
yof
the
18co
rres
pond
ing
indi
vidu
ales
timat
esar
esi
gnifi
cant
atth
e5%
(10%
)co
nfide
nce
leve
l.W
here
p-va
lues
are
noti
nsq
uare
brac
kets
,the
null
hypo
thes
isis
that
the
resp
ectiv
eSR
orsk
ewne
ssdo
esno
texc
eed
the
one
ofth
eSC
trad
e.W
here
p-va
lues
are
insq
uare
brac
kets
,the
null
isth
atth
eSR
orsk
ewne
ssof
aG
1-G
5tr
ade
orG
Ctr
ade
does
note
xcee
dth
atof
the
corr
espo
ndin
gB
1-B
5tr
ade
orB
Ctr
ade.
The
sam
ple
peri
odis
12/1
984
to6/
2014
(354
mon
ths)
.
avg.
ret.
st.d
ev.
SRp-
val
skew
p-va
l
SC1.
023.
300.
31-0
.22
G1
NZ
D,G
BP,
SEK
,CA
D,U
SD,E
UR
,CH
F1.
673.
290.
510.
020.
070.
02G
2G
BP,
SEK
,CA
D,U
SD,C
HF
1.70
3.47
0.49
0.13
-0.1
70.
41G
3N
ZD
,GB
P,SE
K,U
SD,C
HF
2.49
4.09
0.61
0.01
-0.2
10.
48G
4N
ZD
,GB
P,SE
K,C
AD
,USD
2.22
4.39
0.51
0.12
-0.0
10.
19G
5G
BP,
SEK
,USD
3.97
5.71
0.69
0.03
-0.2
30.
50
B1
AU
D,N
OK
,JPY
0.68
7.50
0.09
[0.0
1]-0
.92
[0.0
1]B
2N
ZD
,AU
D,N
OK
,EU
R,J
PY0.
985.
540.
18[0
.07]
-0.6
0[0
.06]
B3
AU
D,N
OK
,CA
D,E
UR
,JPY
0.21
4.91
0.04
[0.0
1]-0
.85
[0.0
4]B
4A
UD
,NO
K,E
UR
,CH
F,JP
Y0.
284.
960.
06[0
.02]
-0.8
7[0
.04]
B5
NZ
D,A
UD
,NO
K,C
AD
,EU
R,C
HF,
JPY
0.61
4.66
0.13
[0.0
1]-0
.63
[0.1
7]
GC
1.96
4.11
0.47
(6/7
)-0
.33
(0/0
)B
C0.
665.
130.
13[(
8/12
)]-0
.66
[(5/
11)]
40
Tabl
e4
Cur
renc
yH
ML
vs.g
ood
carr
ytr
ades
ascu
rren
cym
arke
tpri
cing
fact
ors
Seve
ralf
acto
rpr
icin
gm
odel
sar
ees
timat
edw
ithG
MM
,as
desc
ribe
din
Sect
ion
IV,o
ver
12/1
984
to12
/201
3.T
he11
test
asse
ts(s
ix”A
ll”an
dfiv
e”D
evel
oped
”in
tere
st-r
ate
sort
edcu
rren
cypo
rtfo
lios
(net
retu
rns)
)an
dth
eR
X(d
olla
r)an
dcu
rren
cyH
ML
(den
oted
”HM
LF
X”)
fact
ors
(”A
ll”ve
rsio
n,ne
t)ar
eas
inL
ustig
etal
.(20
11),
and
kind
lym
ade
avai
labl
eat
Adr
ian
Ver
delh
an’s
web
site
.The
good
carr
ytr
ades
G1-
G5
are
asin
Tabl
e3.
All
mod
els
incl
ude
the
RX
fact
or,a
ndei
ther
the
HM
LF
Xor
ago
odca
rry
trad
e(a
sin
dica
ted
inth
efir
stco
lum
n),o
rbo
th.
The
top
pane
lrep
orts
aver
ages
ofth
e11
annu
aliz
edav
erag
ere
turn
s,tim
e-se
ries
regr
essi
onco
effic
ient
san
dad
just
edR
2 ’s(i
npe
rcen
t).S
tand
ard
erro
rsar
ees
timat
edw
ithG
MM
and
acco
unt
foro
neN
ewey
-Wes
tlag
.T
hefir
st(s
econ
d)nu
mbe
rin
pare
nthe
ses
show
sho
wm
any
ofth
e11
corr
espo
ndin
ges
timat
esar
esi
gnifi
cant
atth
e5%
(10%
)co
nfide
nce
leve
l.T
hebo
ttom
pane
lsho
ws,
for
the
sam
em
odel
s,fa
ctor
risk
pric
esλ
and
SDF
coef
ficie
nts
bw
ithp-
valu
es,a
sw
ella
sp-
valu
esfo
rth
eχ
2st
atis
ticte
stin
gth
atth
epr
icin
ger
rors
are
join
tlyeq
ualt
oze
ro.A
vera
gere
turn
s,α
’san
dλ
’sar
ere
port
edan
nual
ized
and
inpe
rcen
t.β
Goo
d,λ
Goo
dan
db G
ood
refe
rto
the
good
carr
ies
G1-
G5,
assh
own
inth
efir
stco
lum
n.
avg.
ret.
p-va
lα
p-va
lβ
RX
p-va
lβ
HM
LFX
p-va
lβ
Goo
dp-
val
R2
2.33
(3/4
)0.
21(1
/2)
1.12
(11/
11)
-0.0
5(8
/8)
81.1
G1
0.06
(1/1
)1.
11(1
1/11
)-0
.01
(10/
10)
78.3
G2
0.06
(1/1
)1.
11(1
1/11
)-0
.01
(9/1
0)75
.9G
30.
06(0
/1)
1.12
(11/
11)
-0.0
1(9
/9)
77.3
G4
0.01
(1/1
)1.
11(1
1/11
)0.
02(8
/9)
75.3
G5
-0.0
1(0
/3)
1.11
(11/
11)
0.02
(7/7
)74
.0G
10.
15(0
/1)
1.12
(11/
11)
-0.0
6(5
/6)
0.08
(6/8
)83
.0G
20.
16(0
/1)
1.12
(11/
11)
-0.0
5(8
/8)
0.05
(10/
10)
81.9
G3
0.12
(0/1
)1.
11(1
1/11
)-0
.06
(7/8
)0.
07(5
/5)
82.2
G4
0.15
(0/1
)1.
1(1
1/11
)-0
.05
(8/8
)0.
06(5
/6)
82.1
G5
0.13
(0/0
)1.
11(1
1/11
)-0
.05
(8/8
)0.
03(7
/7)
81.6
λR
Xp-
val
λH
MLF
Xp-
val
λG
ood
p-va
lb R
Xp-
val
b HM
LFX
p-va
lb G
ood
p-va
lχ
2 pr.e
rr.
2.29
0.08
4.24
0.03
4.03
0.18
4.75
0.06
0.00
G1
2.13
0.10
2.09
0.02
3.43
0.24
17.8
00.
030.
17G
22.
130.
102.
900.
033.
080.
3022
.47
0.04
0.50
G3
2.12
0.10
2.91
0.02
1.70
0.59
15.9
10.
030.
09G
42.
030.
124.
290.
01-4
.38
0.35
24.9
80.
020.
22G
51.
990.
138.
640.
01-8
.54
0.13
29.8
30.
010.
47G
12.
130.
113.
230.
082.
100.
013.
420.
25-0
.08
0.98
18.0
30.
030.
27G
22.
170.
103.
680.
052.
460.
053.
240.
271.
110.
7118
.10
0.12
0.27
G3
2.11
0.11
3.29
0.07
3.03
0.01
1.49
0.64
-0.5
50.
8617
.45
0.06
0.05
G4
2.09
0.11
3.31
0.07
3.74
0.00
-2.8
50.
501.
310.
6020
.26
0.02
0.24
G5
2.13
0.10
3.88
0.04
6.18
0.01
-4.5
70.
302.
660.
2520
.05
0.01
0.97
41
Tabl
e5
Car
ryre
turn
pred
icta
bilit
yT
his
tabl
esh
ows
resu
ltsfr
omun
ivar
iate
pred
ictiv
ere
gres
sion
sfo
rth
elo
gre
turn
sof
the
SCtr
ade,
the
G1-
G5
and
B1-
B5
trad
es,a
ndth
eG
Can
dB
Ctr
ades
asde
scri
bed
inTa
ble
3.T
heth
ree
pred
icto
rssh
own
inth
efir
stro
wof
the
tabl
eha
vebe
enfo
und
topr
edic
tca
rry
trad
ere
turn
sin
Bak
shi
and
Pana
yoto
v(2
013,
Tabl
e2)
and
Rea
dy,
Rou
ssan
ov,
and
War
d(2
017,
Tabl
e10
),an
dar
ede
sign
edas
inth
ose
stud
ies
(see
also
Sect
ion
D).
The
tabl
edi
spla
ysth
ein
-sam
ple
estim
ates
ofth
epr
edic
tive
slop
eco
effic
ient
sβ
,tw
o-si
ded
p-va
lues
,bas
edon
the
Hod
rick
(199
2)1B
cova
rian
cem
atri
xes
timat
or,
and
adju
sted
R2 ’s
(in
perc
ent)
.N
exta
resh
own
one-
side
dp-
valu
es(d
enot
ed”M
S”)
for
the
MSP
E-a
djus
ted
stat
istic
(Cla
rkan
dW
est(
2007
),se
eal
sofo
otno
te6)
,obt
aine
dw
ithan
expa
ndin
gw
indo
ww
ithin
itial
leng
thof
120
mon
ths.
The
colu
mns
deno
ted
”∆SR
”re
port
am
easu
reof
the
econ
omic
sign
ifica
nce
ofpr
edic
tabi
lity,
base
don
apr
edic
tion-
base
dtr
adin
gst
rate
gy,w
hich
ente
rsin
toa
carr
ytr
ade
atth
een
dof
mon
thto
nly
ifth
etr
ade’
sre
turn
pred
icte
dfo
rmon
tht+
1is
posi
tive
(ifa
nega
tive
retu
rnis
pred
icte
d,th
eca
rry
trad
ere
turn
form
onth
t+1
isze
ro).
Spec
ifica
lly,t
hem
easu
reeq
uals
the
diff
eren
cebe
twee
nth
eSh
arpe
ratio
ofth
epr
edic
tion-
base
dst
rate
gy,i
mpl
emen
ted
with
the
resp
ectiv
esu
bset
ofG
-10
curr
enci
es,a
ndth
eco
rres
pond
ing
unco
nditi
onal
carr
ytr
ade
(usi
ngan
expa
ndin
gw
indo
ww
ithin
itial
leng
thof
120
mon
ths)
.Fo
rth
eG
Can
dB
Ctr
ades
the
tabl
esh
ows
aver
ages
ofth
ere
spec
tive
18pr
edic
tive
slop
eco
effic
ient
sβ
,adj
uste
dR
2 ’s,a
ndch
ange
sin
Shar
pera
tios.
The
first
(sec
ond)
num
beri
npa
rent
hese
ssh
ows
how
man
yof
the
18co
rres
pond
ing
indi
vidu
ales
timat
esar
esi
gnifi
cant
atth
e5%
(10%
)con
fiden
cele
vel.
The
sam
ple
peri
odis
12/1
984
to6/
2014
.C
omm
odity
retu
rns
Cha
nge
inex
chan
gera
tevo
latil
ityC
hang
ein
BD
I
βp-
val
R2
MS
∆SR
βp-
val
R2
MS
∆SR
βp-
val
R2
MS
∆SR
SC0.
015
0.12
0.8
0.45
-0.0
4-0
.005
0.00
2.4
0.01
0.10
0.00
30.
151.
00.
180.
08
G1
0.00
60.
530.
10.
99-0
.04
-0.0
030.
120.
80.
22-0
.04
0.00
30.
160.
90.
210.
10G
20.
013
0.19
0.5
0.85
-0.0
5-0
.003
0.21
0.6
0.30
0.00
0.00
20.
390.
30.
790.
00G
30.
008
0.50
0.1
0.82
0.01
-0.0
050.
051.
60.
13-0
.05
0.00
50.
022.
30.
020.
09G
40.
012
0.52
0.3
0.88
-0.0
3-0
.006
0.04
1.9
0.21
-0.0
30.
002
0.48
0.4
0.71
-0.0
1G
5-0
.005
0.82
0.0
0.94
0.00
-0.0
030.
350.
40.
600.
03-0
.002
0.68
0.1
0.97
-0.0
5
B1
0.06
90.
013.
30.
020.
29-0
.010
0.02
1.7
0.01
0.22
0.01
00.
052.
30.
04-0
.14
B2
0.04
30.
042.
30.
060.
15-0
.008
0.01
2.1
0.01
0.13
0.01
00.
014.
20.
010.
09B
30.
032
0.04
1.6
0.10
-0.0
2-0
.007
0.01
2.3
0.01
0.32
0.00
40.
151.
00.
13-0
.15
B4
0.04
00.
032.
50.
060.
12-0
.007
0.02
1.9
0.02
0.23
0.00
70.
062.
30.
050.
09B
50.
029
0.05
1.4
0.12
-0.0
1-0
.006
0.01
1.9
0.01
0.14
0.00
70.
022.
80.
010.
06
GC
0.01
0(0
/3)
0.5
(0/0
)-0
.04
-0.0
05(1
1/14
)1.
4(3
/5)
0.00
0.00
2(2
/3)
0.8
(2/3
)-0
.02
BC
0.03
3(7
/14)
1.7
(1/9
)0.
03-0
.007
(15/
17)
1.8
(17/
18)
0.17
0.00
8(1
2/14
)2.
9(1
2/15
)0.
07
42
Tabl
e6
Car
ryco
mpo
nent
sin
carr
ytr
ade
retu
rns
Thi
sta
ble
show
sth
eav
erag
ere
turn
s,an
dth
eco
mpo
nent
sof
thes
ere
turn
sdu
eto
carr
yan
dex
chan
gera
tech
ange
s,fo
rth
est
anda
rd(S
C)
trad
e,th
ego
odan
dba
dca
rry
trad
esG
1to
G5
and
B1
toB
5,an
dth
edo
llar
carr
ytr
ade
(DC
)of
Lus
tig,R
ouss
anov
,and
Ver
delh
an(2
014)
,w
hich
we
cons
ider
inm
ore
deta
ilin
Sect
ion
B.T
here
spec
tive
valu
esar
esh
own
annu
aliz
edan
din
perc
ent.
Als
osh
own
are
incu
rly
brac
kets
p-va
lues
from
ate
stfo
req
ualit
yof
the
resp
ectiv
eav
erag
eca
rry
com
pone
ntto
that
ofth
eSC
and
DC
trad
es(N
ewey
-Wes
tsta
ndar
der
rors
,w
ith12
lags
).
SCB
1B
2B
3B
4B
5G
1G
2G
3G
4G
5D
C
avg.
tota
lret
.1.
020.
680.
980.
210.
280.
611.
671.
702.
492.
223.
974.
19av
g.de
prec
.-0
.37
-2.0
2-1
.25
-1.5
8-1
.65
-1.2
80.
260.
500.
781.
012.
763.
27av
g.ca
rry
1.38
2.70
2.23
1.78
1.92
1.88
1.40
1.20
1.70
1.20
1.20
0.92
carr
yto
tota
lret
.1.
363.
982.
278.
466.
963.
080.
840.
700.
680.
540.
300.
22
com
pari
ngca
rry
com
pone
nts
(p-v
alue
s)to
that
ofSC
{0.0
0}{0
.00}
{0.0
0}{0
.00}
{0.0
0}{0
.63}
{0.0
8}{0
.00}
{0.0
4}{0
.32}
{0.0
1}to
that
ofD
C{0
.00}
{0.0
0}{0
.00}
{0.0
0}{0
.00}
{0.0
1}{0
.04}
{0.0
0}{0
.04}
{0.0
1}
Tabl
e7
Unb
iase
dnes
shyp
othe
sisr
egre
ssio
nsfo
rpa
irso
fpro
toty
pica
lcar
rytr
ade
curr
enci
es
Fore
ach
ofth
esi
xcu
rren
cypa
irs
NZ
D/C
HF,
AU
D/C
HF,
NO
K/C
HF,
NZ
D/J
PY,A
UD
/JPY
and
NO
K/J
PYm
onth
lylo
gch
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43
Table 8Dynamic nature of carry tradesThe table characterizes the dynamic behavior of the the standard carry trade (SC), the G1 to G5 and B1to B5 trades, and, in the last row, the dollar carry trade (DC) of Lustig et al. (2014), which we considerin more detail in Section B. The first column of the table shows, for each trade, the time-series averageof the proportion of currencies that change (switch) position, from long to short or vice versa, at eachpoint of time. The average proportion is given in percent. The next two columns show bootstrapped 95%confidence intervals for these average proportions. The column denoted ”days w/o switch” shows theproportion of dates in the sample when not a single currency changed position from short to long or viceversa. Next the table shows the correlation between the proportions for the G1 to G5 and B1 to B5 trades,and those for standard carry (SC). The last three columns aim to compare static and dynamic versionsof our various trades. Static trades have been defined in Hassan and Mano (2015), and, to keep close totheir setup, we use as weights the average forward differentials of the respective currencies over 12/1984-12/1994, demeaned and normalized to have absolute values that sum to one. These weights are kept fixedfor the rest of the sample period for the static trades, without ever re-balancing. Dynamic trades are theusual (dynamically re-balanced) trades, as considered throughout this paper, but with weights again equalto the cross-sectionally demeaned forward differentials, normalized to have absolute values that sum toone. Shown are the ratios between the average returns of the respective static and dynamic carry trade, andthe corresponding Sharpe ratios. Average returns and Sharpe ratios are calculated for the period 12/1994to 6/2014.
switch (%) 95% conf. int. days w/o correl. ratio of static to Sharpe ratios:switch with SC dynamic avg. ret. dynamic static
Figure 1: Good and bad carry trades from sets of five G-10 currencies
Large black dots plot skewness versus Sharpe ratio of all possible 21 carry trades constructed from five G-10 currencies, which include the AUD, CHF and JPY, together with any possible pair from the remainingseven currencies. Circles with no fill plot similarly skewness versus Sharpe ratio for the complementarytrades, each including the five currencies left out of one of the previous 21 trades. For each trade, curren-cies are sorted on their forward differentials (against the USD) at the end of each month over the period12/1984 to 6/2014, and the two currencies with highest differentials are held long over the next month,while the two with the lowest premiums are shorted, all with equal weights. A vertical and horizontallines indicate the Sharpe ratio and return skewness of the standard carry trade (denoted SC), constructedwith all G-10 currencies. Percentage carry trade returns are calculated with spot and forward quotes fromBarclays Bank, available via Datastream, and with transaction costs taken into account.
−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
skew
ness
Sharpe ratio
SC
47
Figure 2: Sharpe ratios versus skewness for the GC and BC sets of 18 carry trades
Circles with no fill plot skewness versus Sharpe ratio for 18 carry trades, each constructed from five of theG-10 currencies, with equal weights. Each of these trades uses the three currencies (GBP, SEK and USD)which are least often excluded by the enhancement rule in Section A and Table 2. These three currenciesare combined with any possible pair of the remaining G-10 currencies, which contains none or only oneof the three most often excluded currencies (AUD, NOK and JPY). Large black dots plot similarly theskewness versus Sharpe ratio for the complementary carry trades, which use the five currencies left outof one of the previous 18 trades. These two sets of 18 trades are denoted in Section D and Table 3 andothers as GC and BC. Shown also is the standard carry trade (SC), denoted by a star. The sample periodis 12/1984 to 6/2014.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
−1
−0.8
−0.6
−0.4
−0.2
0
skew
ness
Sharpe ratio
48
Figure 3: Decomposing carry trade returns
For all trades considered in this paper (18 GC trades, 18 BC trades, as well as G1 and G5, and B1 andB5) the figure plots total average returns (horizontal axis) versus the ratios of average carry to total return(vertical axis). As previously, white (black) dots correspond to good (bad) trades. For visual clarity,four outlier points (all referring to bad carry trades) are not shown on this plot, but their coordinatesare displayed in the top left corner. Also plotted are the standard and dollar carry trades (SC and DC).Horizontal lines correspond to carry-to-return ratio of one (all average return comes from carry alone),and one half (return comes equally from carry and exchange rate changes).
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
carry trade return
carry
to re
turn
ratio
(0.05, 28.94 0.21, 8.46 0.20, 7.02 0.27, 6.96)
G1, G5 and GCB1, B5 and BCStandard carryDollar carry
49
Good Carry, Bad Carry
Online Appendix: Not for Publication
OA-I Symmetry and numeraire neutrality of currency trades
This Appendix explains in detail the distinction between several designs of carry trades. Start with
a set of N currencies, e.g. the G-10 currencies in our case. A currency trading strategy is a mapping
between signals at time t and currency positions taken at this time, whereby positions are defined in terms
of the weights of individual currencies. A trading strategy is formulated relative to a benchmark currency,
i.e. positions are taken relative to a certain currency in the forward market. From this perspective, two
properties seem important:
1. Symmetry: the number of short and long positions and their total weights are equal. A stronger
version of symmetry would also require equal weights of the individual short or long positions.1
2. Numeraire independence: the positions taken in the various currencies are the same, regardless of
which benchmark currency is considered. As a result, only one currency strategy must be defined
for the world at large.
Symmetry and numeraire independence are well-established features of carry trades, and have been both
adopted by recent academic studies, and implemented in investable products (see Table 1). Together, these
properties imply that the trade’s returns will be very similar from any currency perspective. This invariance
follows from the fact that the translation of returns from one currency to another simply introduces cross-
currency risk on currency returns, which is a second order effect. Conversely, if the ranking of a currency
or the signal depends in any way on the identity of the benchmark currency, then defining the same strategy
from another currency perspective will yield different currency positions and different currency weights,
and this can result in quite different returns. A well-known example is the asymmetric carry strategy in
Burnside, Eichenbaum, Kleshchelski, and Rebelo (2011), which has been shown in Daniel, Hodrick, and
1However, if weights are defined relative to a benchmark currency (e.g., based on forward differentials), they may differ onthe long and short end, creating weight asymmetry. This would cause the trade to be numeraire-dependent.
1
Lu (2017) to produce very different (and worse) returns from other, non-USD currency perspectives. In
fact, the USD-based version of this strategy is successful (at least partly) due to is its implicit exposure to
a dollar-centric currency strategy, the ”dollar carry” trade of Lustig, Roussanov, and Verdelhan (2014).2
We now formally show that symmetric, numeraire-independent strategies have largely equivalent re-
turns across the world. Suppose first that the USD is the benchmark currency and define the weight of
currency i as wi. Spot and forward exchange rates are quoted here as USD per one unit of a foreign cur-
rency (reversing the notation from Section II above), and denoted as Sit and F i
t . The return of a US-based
currency trading strategy over the interval t to t +1 is:
(OA-1) rUSDt+1 =
N
∑i=1
wi[Sit+1/F i
t −1].
If the strategy is numeraire independent, the weights wi are identical for all currency perspectives. For
example, if the trading strategy is based on interest rate signals, these signals should be independent of the
benchmark currency.
Defining such a strategy relative, say, to the Japanese yen, with yen exchange rates denoted by Sit and
F it (JPY per one unit of currency i), its return (in yen) is:
(OA-2) rJPYt+1 =
N
∑i=1
wi[Sit+1/F i
t−1] =N
∑i=1
wiSit+1/F i
t−N
∑i=1
wi
With symmetric strategies the weights sum to zero, hence the last term cancels, and we are left with
rJPYt+1 = ∑
Ni=1 wiS
it+1/F i
t . By triangular arbitrage and symmetry, we can further derive:
rJPYt+1 = [rUSD
t+1 +N
∑i=1
wi]∗SUSDt+1 /FUSD
t = rUSDt+1 ∗FJPY
t /SJPYt+1 .(OA-3)
2The dollar carry weights are 1/(N-1) and are all either positive or negative depending on the average interest rate of theUSD relative to other currencies. The weight on the USD itself is zero. The strategy is thus very asymmetric and yields entirelydifferent results for other currency perspectives. That is, a ”British pound carry” or ”Swiss franc carry” need not be anythinglike dollar carry. Of course, dollar-centric strategies are of interest because of the importance of the dollar in internationalfinance.
2
Cross-currency risk could drive, in principle, a wedge between the two currency perspectives, but in
practice the returns and their properties will be rather similar (barring significant differences in transaction
costs), because the forward to spot ratio in (OA-3) is close to one, and applies to returns. We have verified
that standard carry strategies (as per our definition in Section II) yield very similar returns from any
currency perspective.
It is instructive to repeat the previous calculation, but for log returns. In this case:
rJPYt+1 =
N
∑i=1
wi log(
Sit+1/F i
t
)=
N
∑i=1
wi log
(Si
t+1
F it
SUSDt+1
FUSDt
)
=N
∑i=1
wi log(Sit+1/F i
t )+N
∑i=1
wi log(SUSDt+1 /FUSD
t )
=N
∑i=1
wi log(Sit+1/F i
t )+ log(SUSDt+1 /FUSD
t )N
∑i=1
wi
=N
∑i=1
wi log(Sit+1/F i
t )+0 = rUSDt+1 ,(OA-4)
and therefore the log returns of symmetric, numeraire-independent trades are identical from any perspec-
tive; the differences between their percentage returns from different perspectives are of second order.
In sum, a symmetric carry trade, for any benchmark currency has similar returns for investors across
the world. However, symmetry is not a sufficient condition for numeraire independence. It is important to
emphasize this point, because a number of recent articles have considered ”currency-neutral” symmetric
strategies, where no position is taken with respect the benchmark currency itself, or in other words, the
weight assigned to the benchmark currency is always zero (this is implicitly true also for dollar carry).
Let’s examine, following Daniel, Hodrick, and Lu (2017), a ”dollar-neutral” carry trade with weights
wi = 1/(N − 1) if the interest rate of currency i is in top half of the interest rates of the given set of
currencies, and wi =−1/(N−1) otherwise (if N-1 is odd, the currency with the median interest rate is left
out of the trade). This strategy is clearly symmetric. However, it is not numeraire independent because if
3
we define it relative to another benchmark currency, say the yen, the weight function of this ”yen-neutral”
trade will change, with now non-zero weights on the USD and zero weights on the JPY. Therefore, such
”currency-neutral” trades will produce different returns for different benchmark currencies, going beyond
the differences induced by cross-currency risk.
We recognize that some numeraire-dependent strategies are of obvious interest, but care must be taken
to define them in an international context. For example, the HML factor, introduced by Lustig, Roussanov,
and Verdelhan (2011, 2014) is a carry trade which is symmetric, but not numeraire-independent as it goes
long (short) an extreme portfolio based on an interest rate ranking (as the DB strategy does), but excludes
the USD from any portfolio. This dollar neutrality makes the trade numeraire-dependent. Of course,
when such a trade is defined for benchmark currencies with non-extreme interest rates, it should often
yield similar returns across the different country perspectives.
Our preference for using symmetric, numeraire-independent carry trades is consistent with the best
known investable indices, such as the Deutsche Bank (DB) Harvest Indexes. The DB strategy goes long
(short) the G-10 currencies with the three highest (lowest) interest rates. Importantly, when the USD
interest rate is among the top or bottom three, part of the trade automatically gets a zero return, because a
position in the benchmark currency itself is taken, and hence the trade is not dollar-neutral. However, it is
symmetric and numeraire-independent, which is an advantage for a global currency trading strategy, and
may also be an advantage for a global risk pricing factor. In the trades that we consider, all participating
currencies are given a non-zero weight, including the benchmark currency, which by design yields a zero
return, whether it is held long or short.
Another way to see the fundamental difference between asymmetric, numeraire-dependent trades on
the one hand, and numeraire-independent strategies on the other is to examine what would happen if, say, a
yen-based investor would try to mimic, for example, dollar carry by taking exactly the same positions, but
4
relative to the yen. That is, she will go long or short in all the currencies (including the yen) as dollar carry
does, thus keeping the same weight function as in the original dollar trade, but for a different benchmark
currency. This strategy would yield quite different returns as it would face full cross-currency risk, and
not just profit and loss currency risk.
With the previous notation: rJPYt+1 = [rUSD
t+1 +∑Ni=1 wi] ∗ SUSD
t+1 /FUSDt −∑
Ni=1 wi. Since for dollar carry
these weights add up to one, and not zero as in a symmetric trade, the yen-based return is now:
(OA-5) rJPYt+1 = [rUSD
t+1 +1]∗SUSDt+1 /FUSD
t −1 = rUSDt+1 FJPY
t /SJPYt+1 +[FJPY
t /SJPYt+1 −1],
which, compared to the expression in (OA-3), adds a second return term that can well be of similar or
even larger magnitude than the first term.
OA-II Tests for differences in Sharpe ratios and return skewness
Sharpe ratios
The statistical significance of the differences between the Sharpe ratio or skewness of the SC trade
and those of trades from subsets is evaluated using bootstrap tests that follow Ledoit and Wolf (2008) or
Annaert, Van Osselaer, and Verstraete (2009). Skewness difference can be tested in a ”direct” bootstrap
that resamples from a distribution which respects the null hypothesis of no difference. In the case of
Sharpe ratios, their difference does not easily admit such a distribution, hence the approach followed is
”indirect” and resamples from the observed data. A version of this approach to comparing Sharpe ratios
has been applied recently, among others, in DeMiguel, Nogales, and Uppal (2014).
In implementing the test for a difference between Sharpe ratios, we depart in two minor ways from
Ledoit and Wolf (2008). First, we only consider the i.i.d. case (their Section 3.2.1). We have verified
that our carry trade return series have insignificant autocorrelations for lags up to 10. Furthermore, the
suggested block size selection procedure (their Algorithm 3.1) results consistently in a selected block
5
length of one, when using our data. Second, we consider one-sided bootstrap confidence intervals and
p-values, since our null hypothesis is that carry trades obtained with the enhancement rule do not improve
on the Sharpe ratio of the SC trade. We modify accordingly their equation (7).
Following the notation in Ledoit and Wolf (2008), let µS and µB denote the sample average returns of
a carry trade from some subset of the G-10 currencies and the SC trade, respectively, while γS and γB are
the sample second moments (uncentered) of the returns of these trades. Let also v = (µS, µB, γS, γB), and
assume that√
T (v−v) d→ (0,Ψ), where v is the population counterpart, T is sample length and Ψ is some
symmetric positive-definite matrix. The latter assumption holds under mild conditions. For the sample
difference ∆ between the Sharpe ratios of the carry trade from a subset of the G-10 currencies and the SC
trade, and the deviation of this sample difference from the population value ∆, one can write
∆ = f (v) =µS
γS− µ2S− µB
γB− µ2B
and√
T (∆−∆)d→ (0,∇′ f (v)Ψ∇ f (v)),(OA-6)
where ∇′ f ((a,b,c,d)) =(
c(c−a2)1.5 ,− d
(d−b2)1.5 ,− a2(c−a2)1.5 ,
b2(d−b2)1.5
)and (a, b, c, d) represent the ele-
ments in v. If Ψ is a consistent estimator of Ψ, then the standard error of ∆ is given by
(OA-7) s(∆) =
√∇′ f (v)Ψ∇ f (v)
T.
To test the null hypothesis ∆ ≤ 0, we bootstrap the returns of the two carry trades that are compared,
and consider the studentized random variable L= ∆∗−∆
s(∆∗) , where ∆∗ is a difference in Sharpe ratios computed
with bootstrapped returns, and s(∆∗) is the corresponding standard error. Even though we bootstrap ”under
the alternative”, this procedure generates meaningful sampling variation under the null of no difference
between Sharpe ratios. Given the lack of autocorrelation in the carry trade return series, as noted above,
we use an i.i.d. bootstrap (5000 samples, with replacement and pairwise, to preserve a possible cross-
sectional correlation between the returns of the two carry trades). A p-value for the null is calculated as
6
the proportion of bootstrapped series for which:
(OA-8) ∆−L = ∆+∆−∆∗
s(∆∗)s(∆)≤ 0,
similar to equation (7) in Ledoit and Wolf (2008). These p-values are reported in Tables 2 and 3.
Skewness
To test for a difference in skewness, Annaert, Van Osselaer, and Verstraete (2009, page 277) first ”sym-
metrize” the compared return series, by appending to them the mirror images of the original observations
in terms of distance to the average return. The skewness (as well as any odd central moment) of these
modified returns is thus zero, and a bootstrap that resamples from them conforms to the null of no differ-
ence in skewness. Given that autocorrelation does not seem to be an issue in our series, we draw pairwise
from the modified series of the compared returns, and compute the p-value as the percentage of draws that
yield higher improvement on the benchmark skewness than that observed in the data. All bootstraps are
performed with 5000 draws.
OA-III Differences in Sharpe ratios - accommodating the selection
The enhancement procedure described in Section III introduces a possible selection bias, which is not
accounted for by the bootstrap-based test described above, following Ledoit and Wolf (2008). To address
this issue, we suggest an alternative approach, and instead of bootstrapping the actual carry returns, we
adopt the following randomization procedure:
• at the end of month t keep the interest rate differentials as in the data, but assign to each of them at
random any of the ten returns for the following month t +1.
• to each of these ten returns for month t +1 add the same constant ct+1. We call the returns obtained
in this way ”randomized” returns.
7
• the constant ct+1 can be positive or negative, and is chosen so that a carry trade that uses all ten
”randomized” returns would have exactly the same return as the actual SC trade for month t + 1.
Such a carry trade would choose the currencies to be long or short exactly as the SC trade, based on
sorting the same interest rate differentials.
• do this for all months in the sample, and repeat 1000 times, to obtain 1000 sets of ten ”random-
ized” return series, that correspond to the actual interest rate differentials. Given the large number
permutations of ten numbers, we do not bootstrap in addition the interest rate differentials.
• note that the constants ct+1 are different for different months, and that each ”randomized” return
corresponding to a particular interest rate differential is potentially very different from the actual
one. This approach may associate, for example, the JPY returns predominantly with the highest
interest rates in some randomization trials. However, the returns for each month, and hence the
Sharpe ratios of the carry trades with ten currencies (all 1000 with ”randomized” returns and the
actual SC trade) are exactly the same.
• on each of the 1000 sets of 10 time series reproduce the enhancement procedure described in Section
B. Based on the order of exclusion obtained from this procedure, identify for each of the 1000 sets
the currencies that would enter ”good” and ”bad” carry trades.
• in the full sample period construct trades with the least excluded three, five or seven currencies,
corresponding to our G1-G5 trades, and similarly with the most often excluded three, five or seven
currencies, corresponding to our B1-B5 trades.
For each of 1000 sets of 10 series of randomized carry trade returns, ∆∗ denotes the difference between
the annualized Sharpe ratio of a good carry trade (from three, five or seven currencies), constructed from
this set following the enhancement procedure, and the SC trade or the corresponding bad trade. As in
8
Appendix OA-II, ∆ denotes the sample difference between the annualized Sharpe ratio of a good carry
trade and the SC trade or the corresponding bad trade. We now show ∆ for each good carry trade, the
average of the 1000 ∆∗’s for trades from as many currencies as the good trade on the same line, and the
proportion of such ∆∗’s exceeding ∆.
Good trades vs. SC Good vs. bad trades
∆ avg. ∆∗ % ∆∗ > ∆ ∆ avg. ∆∗ % ∆∗ > ∆
G1 0.20 0.14 0.21 0.42 0.42 0.50
G2 0.18 0.16 0.43 0.31 0.41 0.71
G3 0.30 0.16 0.09 0.57 0.41 0.19
G4 0.20 0.16 0.37 0.45 0.41 0.41
G5 0.39 0.14 0.02 0.56 0.32 0.05
There is substantial bias in the comparison between the G1-G5 carry trades with the SC trade, with
the selection procedure adding 14% (for G1 and G5) or 16% (for G2 to G4) to the annualized Sharpe
ratio. Yet, in every case the observed increases in the Sharpe ratio (denoted by ∆) are even higher, and for
two out of the five good trades the observed Sharpe ratio is in the 10% right tail of the distribution of the
Sharpe ratios obtained under the selection procedure using the randomized (scrambled) currency returns.
When comparing the G1-G5 carry trades to the corresponding B1-B5 trades, the bias is relatively more
important, and in fact at least as large as the observed difference in Sharpe ratios for the G1 and G2 trades.
Only the G5 versus B5 comparison yields a Sharpe ratio of a good trade in the right tail (5.3%) of the
corresponding distribution under scrambled currency returns.
Of course, these observations alone do not constitute a proper test, since the randomization procedure
also can change the variability of the returns, and proper testing requires the use of a pivotal test statis-
tic, such as a t-statistic. To create a proper test statistic, we modify the procedure in Ledoit and Wolf
9
(2008) by bias-correcting our sample Sharpe ratios, and using t-statistics from the empirical distribution
as in Appendix OA-II. The results, which also reproduce the relevant portion from Table 3, to facilitate
comparison are as follows:
Good trades vs. SC Good vs. bad trades
av.ret std. SR bstrp. rand. av.ret std. SR bstrp. rand.
Figure 4: Average vs. model-based expected returns
Circles with no fill (large black dots) plot model-based expected monthly returns versus average monthlyreturns (annualized and in percent) for the GC (BC) set of 18 carry trades, as described in Table 3. Themodel based returns refer to the three-factor model with a market factor (MKT), an equity volatility factor(EqVol) and the product of MKT and EqVol, and are estimated, for each trade, as the product of its time-series slope estimates (β) with respect to the factors in the model, and the corresponding estimates of thefactor risk prices λ, as shown in Table OA-2. The bottom right corner of each plot shows the R2 obtainedin regressing average returns on model-based returns (with a constant). The sample period is 12/1984 to12/2013.
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